Master File

User Inputs

output.var = params$output.var 

transform.abs = FALSE
log.pred = params$log.pred
norm.pred = FALSE
algo.forward.caret = params$algo.forward.caret
algo.backward.caret = params$algo.backward.caret
algo.stepwise.caret = params$algo.stepwise.caret
algo.LASSO.caret = params$algo.LASSO.caret
algo.LARS.caret = params$algo.LARS.caret
message("Parameters used for training/prediction: ")

## Parameters used for training/prediction:

str(params)

## List of 7
##  $ output.var         : chr "y3"
##  $ log.pred           : logi TRUE
##  $ algo.forward.caret : logi TRUE
##  $ algo.backward.caret: logi TRUE
##  $ algo.stepwise.caret: logi TRUE
##  $ algo.LASSO.caret   : logi TRUE
##  $ algo.LARS.caret    : logi TRUE

# Setup Labels
output.var.tr = if (log.pred == TRUE)  paste0(output.var,'.log') else  output.var.tr = output.var

Loading Data

feat  = read.csv('../../Data/features_highprec.csv')
labels = read.csv('../../Data/labels.csv')
predictors = names(dplyr::select(feat,-JobName))
data.ori = inner_join(feat,labels,by='JobName')
#data.ori = inner_join(feat,select_at(labels,c('JobName',output.var)),by='JobName')

Data validation

cc  = complete.cases(data.ori)
data.notComplete = data.ori[! cc,]
data = data.ori[cc,] %>% select_at(c(predictors,output.var,'JobName'))
message('Original cases: ',nrow(data.ori))

## Original cases: 10000

message('Non-Complete cases: ',nrow(data.notComplete))

## Non-Complete cases: 3020

message('Complete cases: ',nrow(data))

## Complete cases: 6980

Output Variable

The Output Variable y3 shows right skewness, so will proceed with a log transofrmation

Histogram

ggplot(gather(select_at(data,output.var)), aes(value)) + 
  geom_histogram(aes(y=..density..),bins = 50,fill='light blue') + 
  geom_density() + 
  facet_wrap(~key, scales = 'free',ncol=4)

QQPlot

ggplot(gather(select_at(data,output.var)), aes(sample=value)) + 
  stat_qq() + 
  facet_wrap(~key, scales = 'free',ncol=4)

Trasformation of Output Variable from y3 to y3.log

if(log.pred==TRUE) data[[output.var.tr]] = log(data[[output.var]],10) else
  data[[output.var.tr]] = data[[output.var]]

ggplot(gather(select_at(data,c(output.var,output.var.tr))), aes(value)) + 
  geom_histogram(aes(y=..density..),bins = 50,fill='light blue') + 
  geom_density() + 
  facet_wrap(~key, scales = 'free',ncol=2)

ggplot(gather(select_at(data,c(output.var,output.var.tr))), aes(sample=value)) + 
  stat_qq() + 
  facet_wrap(~key, scales = 'free',ncol=4)

Best Normalizator y3

Normalization of y3 using bestNormalize package. (suggested orderNorm) This is cool, but I think is too far for the objective of the project

t=bestNormalize::bestNormalize(data[[output.var]])

t

## Best Normalizing transformation with 6980 Observations
##  Estimated Normality Statistics (Pearson P / df, lower => more normal):
##  - No transform: 2.8788 
##  - Box-Cox: 1.3966 
##  - Log_b(x+a): 1.9242 
##  - sqrt(x+a): 2.3331 
##  - exp(x): 748.7106 
##  - arcsinh(x): 1.9245 
##  - Yeo-Johnson: 1.1012 
##  - orderNorm: 1.0521 
## Estimation method: Out-of-sample via CV with 10 folds and 5 repeats
##  
## Based off these, bestNormalize chose:
## orderNorm Transformation with 6980 nonmissing obs and no ties 
##  - Original quantiles:
##      0%     25%     50%     75%    100% 
##  95.913 118.289 124.030 131.059 193.726

qqnorm(data[[output.var]])

qqnorm(predict(t))

orderNorm() is a rank-based procedure by which the values of a vector are mapped to their percentile, which is then mapped to the same percentile of the normal distribution. Without the presence of ties, this essentially guarantees that the transformation leads to a uniform distribution

Predictors

All predictors show a Fat-Tail situation, where the two tails are very tall, and a low distribution around the mean. The orderNorm transromation can help (see [Best Normalizator] section)

Interesting Predictors

Histograms

cols = c('x11','x18','stat98','x7','stat110')
ggplot(gather(select_at(data,cols)), aes(value)) + 
  geom_histogram(aes(y=..density..),bins = 50,fill='light blue') + 
  geom_density() + 
  facet_wrap(~key, scales = 'free',ncol=3)

# ggplot(gather(select_at(data,cols)), aes(sample=value)) + 
#   stat_qq()+
#   facet_wrap(~key, scales = 'free',ncol=2)

lapply(select_at(data,cols),summary)

## $x11
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 9.000e-08 9.494e-08 1.001e-07 1.001e-07 1.052e-07 1.100e-07 
## 
## $x18
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.500   3.147   4.769   4.772   6.418   7.999 
## 
## $stat98
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -2.998619 -1.551882 -0.015993 -0.005946  1.528405  2.999499 
## 
## $x7
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.700   1.266   1.854   1.852   2.446   3.000 
## 
## $stat110
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -2.999543 -1.496865 -0.002193 -0.004129  1.504273  2.999563

Scatter plot vs. output variable **y3.log

d = gather(dplyr::select_at(data,c(cols,output.var.tr)),key=target,value=value,-!!output.var.tr)
ggplot(data=d, aes_string(x='value',y=output.var.tr)) + 
  geom_point(color='light green',alpha=0.5) + 
  geom_smooth() + 
  facet_wrap(~target, scales = 'free',ncol=3)

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

All Predictors

Histograms

All indicators have a strong indication of Fat-Tails

ggplot(gather(select_at(data,predictors)), aes(value)) + 
  geom_histogram(aes(y=..density..),bins = 50,fill='light blue') + 
  geom_density() + 
  facet_wrap(~key, scales = 'free',ncol=4)

Correlations

With Output Variable

#chart.Correlation(select(data,-JobName),  pch=21)
t=as.data.frame(round(cor(dplyr::select(data,-one_of(output.var.tr,'JobName'))
                          ,select_at(data,output.var.tr)),4))  %>%
  rownames_to_column(var='variable') %>% filter(variable != !!output.var) %>% arrange(-y3.log)
#DT::datatable(t)
message("Top Positive")

## Top Positive

kable(head(arrange(t,desc(y3.log)),20))

variable	y3.log
x18	0.3120
x7	0.2091
stat98	0.1784
x9	0.1127
x17	0.0611
x16	0.0489
x10	0.0472
x21	0.0412
x11	0.0322
x8	0.0318
stat156	0.0287
stat23	0.0234
stat100	0.0206
stat144	0.0203
stat59	0.0202
stat60	0.0199
stat195	0.0199
stat141	0.0194
stat73	0.0192
stat197	0.0185

message("Top Negative")

## Top Negative

kable(head(arrange(t,y3.log),20))

variable	y3.log
stat110	-0.1594
x4	-0.0603
stat13	-0.0345
stat41	-0.0345
stat14	-0.0317
stat149	-0.0309
stat113	-0.0279
stat4	-0.0248
stat106	-0.0236
stat146	-0.0236
stat186	-0.0217
stat91	-0.0210
stat214	-0.0209
stat5	-0.0207
stat22	-0.0202
stat39	-0.0202
stat175	-0.0194
stat187	-0.0193
stat128	-0.0192
stat37	-0.0191

Between All Variables

#chart.Correlation(select(data,-JobName),  pch=21)
t=as.data.frame(round(cor(dplyr::select(data,-one_of('JobName'))),4))
#DT::datatable(t,options=list(scrollX=T))
message("Showing only 10 variables")

## Showing only 10 variables

kable(t[1:10,1:10])

	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10
x1	1.0000	0.0034	-0.0028	0.0085	0.0068	0.0159	0.0264	-0.0012	0.0142	0.0013
x2	0.0034	1.0000	-0.0057	0.0004	-0.0094	-0.0101	0.0089	0.0078	0.0049	-0.0214
x3	-0.0028	-0.0057	1.0000	0.0029	0.0046	0.0006	-0.0105	-0.0002	0.0167	-0.0137
x4	0.0085	0.0004	0.0029	1.0000	-0.0059	0.0104	0.0098	0.0053	0.0061	-0.0023
x5	0.0068	-0.0094	0.0046	-0.0059	1.0000	0.0016	-0.0027	0.0081	0.0259	-0.0081
x6	0.0159	-0.0101	0.0006	0.0104	0.0016	1.0000	0.0200	-0.0157	0.0117	-0.0072
x7	0.0264	0.0089	-0.0105	0.0098	-0.0027	0.0200	1.0000	-0.0018	-0.0069	-0.0221
x8	-0.0012	0.0078	-0.0002	0.0053	0.0081	-0.0157	-0.0018	1.0000	0.0142	-0.0004
x9	0.0142	0.0049	0.0167	0.0061	0.0259	0.0117	-0.0069	0.0142	1.0000	0.0149
x10	0.0013	-0.0214	-0.0137	-0.0023	-0.0081	-0.0072	-0.0221	-0.0004	0.0149	1.0000

Scatter Plots with Output Variable

Scatter plots with all predictors and the output variable (y3.log)

d = gather(dplyr::select_at(data,c(predictors,output.var.tr)),key=target,value=value,-!!output.var.tr)
ggplot(data=d, aes_string(x='value',y=output.var.tr)) + 
  geom_point(color='light blue',alpha=0.5) + 
  geom_smooth() + 
  facet_wrap(~target, scales = 'free',ncol=4)

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

Multicollinearity - VIF

No Multicollinearity among predictors

Showing Top predictor by VIF Value

vifDF = usdm::vif(select_at(data,predictors)) %>% arrange(desc(VIF))
head(vifDF,15)

##    Variables      VIF
## 1    stat174 1.064302
## 2    stat150 1.062254
## 3    stat147 1.059959
## 4     stat46 1.059717
## 5    stat178 1.059640
## 6     stat35 1.058976
## 7    stat169 1.058927
## 8     stat19 1.058803
## 9      stat6 1.058525
## 10    stat95 1.058466
## 11    stat72 1.058419
## 12        x3 1.058266
## 13   stat202 1.058198
## 14   stat205 1.058128
## 15   stat128 1.058010

Feature Eng

• Square Root transformation for x18

data.tr=data %>%
  mutate(x18.sqrt = sqrt(x18)) 
cols=c('x18','x18.sqrt')

Comparing Pre and Post Transformation Density Plots

# ggplot(gather(select_at(data.tr,cols)), aes(value)) + 
#   geom_histogram(aes(y=..density..),bins = 50,fill='light blue') + 
#   geom_density() + 
#   facet_wrap(~key, scales = 'free',ncol=4)

d = gather(dplyr::select_at(data.tr,c(cols,output.var.tr)),key=target,value=value,-!!output.var.tr)
ggplot(data=d, aes_string(x='value',y=output.var.tr)) + 
  geom_point(color='light blue',alpha=0.5) + 
  geom_smooth() + 
  facet_wrap(~target, scales = 'free',ncol=4)

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

#removing unwanted variables
data.tr=data.tr %>%
  dplyr::select_at(names(data.tr)[! names(data.tr) %in% c('x18','y3','JobName')])

data=data.tr
label.names=output.var.tr

Modeling

Train Test Split

data = data[sample(nrow(data)),] # randomly shuffle data
split = sample.split(data[,label.names], SplitRatio = 0.8)

data.train = subset(data, split == TRUE)
data.test = subset(data, split == FALSE)

Common Functions

plot.diagnostics <-  function(model, train) {
  plot(model)
  
  residuals = resid(model) # Plotted above in plot(lm.out)
  r.standard = rstandard(model)
  r.student = rstudent(model)

  plot(predict(model,train),r.student,
      ylab="Student Residuals", xlab="Predicted Values", 
      main="Student Residual Plot") 
  abline(0, 0)
  
  plot(predict(model, train),r.standard,
      ylab="Standard Residuals", xlab="Predicted Values", 
      main="Standard Residual Plot") 
  abline(0, 0)
  abline(2, 0)
  abline(-2, 0)
  
  # Histogram
  hist(r.student, freq=FALSE, main="Distribution of Studentized Residuals", 
  xlab="Studentized Residuals", ylab="Density", ylim=c(0,0.5))

  # Create range of x-values for normal curve
  xfit <- seq(min(r.student)-1, max(r.student)+1, length=40)

  # Generate values from the normal distribution at the specified values
  yfit <- (dnorm(xfit))

  # Add the normal curve
  lines(xfit, yfit, ylim=c(0,0.5))
  
  # http://www.stat.columbia.edu/~martin/W2024/R7.pdf
  # Influential plots
  inf.meas = influence.measures(model)
  # print (summary(inf.meas)) # too much data
  
  # Leverage plot
  lev = hat(model.matrix(model))
  plot(lev, ylab = 'Leverage - check')
  
  # Cook's Distance
  cd = cooks.distance(model)
  plot(cd,ylab="Cooks distances")
  abline(4/nrow(train),0)
  abline(1,0)
  
  print (paste("Number of data points that have Cook's D > 4/n: ", length(cd[cd > 4/nrow(train)]), sep = "")) 
  print (paste("Number of data points that have Cook's D > 1: ", length(cd[cd > 1]), sep = "")) 
  return(cd)
}

# function to set up random seeds
# Based on http://jaehyeon-kim.github.io/2015/05/Setup-Random-Seeds-on-Caret-Package.html 
setCaretSeeds <- function(method = "cv", numbers = 1, repeats = 1, tunes = NULL, seed = 1701) {
  #B is the number of resamples and integer vector of M (numbers + tune length if any)
  B <- if (method == "cv") numbers
  else if(method == "repeatedcv") numbers * repeats
  else NULL
  if(is.null(length)) {
    seeds <- NULL
  } else {
    set.seed(seed = seed)
    seeds <- vector(mode = "list", length = B)
    seeds <- lapply(seeds, function(x) sample.int(n = 1000000
                                                  , size = numbers + ifelse(is.null(tunes), 0, tunes)))
    seeds[[length(seeds) + 1]] <- sample.int(n = 1000000, size = 1)
  }
  # return seeds
  seeds
}

train.caret.glmselect = function(formula, data, method
                                 ,subopt = NULL, feature.names
                                 , train.control = NULL, tune.grid = NULL, pre.proc = NULL){
  
  if(is.null(train.control)){
    train.control <- trainControl(method = "cv"
                              ,number = 10
                              ,seeds = setCaretSeeds(method = "cv"
                                                     , numbers = 10
                                                     , seed = 1701)
                              ,search = "grid"
                              ,verboseIter = TRUE
                              ,allowParallel = TRUE
                              )
  }
  
  if(is.null(tune.grid)){
    if (method == 'leapForward' | method == 'leapBackward' | method == 'leapSeq'){
      tune.grid = data.frame(nvmax = 1:length(feature.names))
    }
    if (method == 'glmnet' && subopt == 'LASSO'){
      # Will only show 1 Lambda value during training, but that is OK
      # https://stackoverflow.com/questions/47526544/why-need-to-tune-lambda-with-carettrain-method-glmnet-and-cv-glmnet
      # Another option for LASSO is this: https://github.com/topepo/caret/blob/master/RegressionTests/Code/lasso.R
      lambda = 10^seq(-2,0, length =100)
      alpha = c(1)
      tune.grid = expand.grid(alpha = alpha,lambda = lambda)
    }
    if (method == 'lars'){
      # https://github.com/topepo/caret/blob/master/RegressionTests/Code/lars.R
      fraction = seq(0, 1, length = 100)
      tune.grid = expand.grid(fraction = fraction)
      pre.proc = c("center", "scale") 
    }
  }
  
  # http://sshaikh.org/2015/05/06/parallelize-machine-learning-in-r-with-multi-core-cpus/
  cl <- makeCluster(detectCores()*0.75) # use 75% of cores only, leave rest for other tasks
  registerDoParallel(cl)

  set.seed(1) 
  # note that the seed has to actually be set just before this function is called
  # settign is above just not ensure reproducibility for some reason
  model.caret <- caret::train(formula
                              , data = data
                              , method = method
                              , tuneGrid = tune.grid
                              , trControl = train.control
                              , preProc = pre.proc
                              )
  
  stopCluster(cl)
  registerDoSEQ() # register sequential engine in case you are not using this function anymore
  
  if (method == 'leapForward' | method == 'leapBackward' | method == 'leapSeq'){
    print("All models results")
    print(model.caret$results) # all model results
    print("Best Model")
    print(model.caret$bestTune) # best model
    model = model.caret$finalModel

    # Metrics Plot 
    dataPlot = model.caret$results %>%
      gather(key='metric',value='value',-nvmax) %>%
      dplyr::filter(metric %in% c('MAE','RMSE','Rsquared'))
    metricsPlot = ggplot(data=dataPlot,aes(x=nvmax,y=value) ) +
      geom_line(color='lightblue4') +
      geom_point(color='blue',alpha=0.7,size=.9) +
      facet_wrap(~metric,ncol=4,scales='free_y')+
      theme_light()
    plot(metricsPlot)
    
    # Residuals Plot
    # leap function does not support studentized residuals
    dataPlot=data.frame(pred=predict(model.caret,data),res=resid(model.caret))
    residPlot = ggplot(dataPlot,aes(x=pred,y=res)) +
      geom_point(color='light blue',alpha=0.7) +
      geom_smooth(method="lm")+
      theme_light()
    plot(residPlot)
   
    residHistogram = ggplot(dataPlot,aes(x=res)) +
      geom_histogram(aes(y=..density..),fill='light blue',alpha=1) +
      geom_density(color='lightblue4') + 
      theme_light()
    plot(residHistogram)
    id = rownames(model.caret$bestTune)    
    # Provides the coefficients of the best model
    # regsubsets doens return a full model (see documentation of regsubset), so we need to recalcualte themodel
    # https://stackoverflow.com/questions/13063762/how-to-obtain-a-lm-object-from-regsubsets
    print("Coefficients of final model:")
    coefs <- coef(model, id=id)
    #calculate the model to the the coef intervals
    nams <- names(coefs)
    nams <- nams[!nams %in% "(Intercept)"]
    response <-  as.character(formula[[2]])
    form <- as.formula(paste(response, paste(nams, collapse = " + "), sep = " ~ "))
    mod <- lm(form, data = data)
    #coefs
    #coef(mod)
    print(car::Confint(mod))
    return(list(model = model,id = id, residPlot = residPlot, residHistogram=residHistogram
                ,modelLM=mod))
  }
  if (method == 'glmnet' && subopt == 'LASSO'){
    print(model.caret)
    print(plot(model.caret))
    print(model.caret$bestTune)
    
    print(model.caret$results)
    model=model.caret$finalModel
    # Metrics Plot 
    dataPlot = model.caret$results %>%
      gather(key='metric',value='value',-lambda) %>%
      dplyr::filter(metric %in% c('MAE','RMSE','Rsquared'))
    metricsPlot = ggplot(data=dataPlot,aes(x=lambda,y=value) ) +
      geom_line(color='lightblue4') +
      geom_point(color='blue',alpha=0.7,size=.9) +
      facet_wrap(~metric,ncol=4,scales='free_y')+
      theme_light()
    plot(metricsPlot)
    
    # Residuals Plot 
    dataPlot=data.frame(pred=predict(model.caret,data),res=resid(model.caret))
    residPlot = ggplot(dataPlot,aes(x=pred,y=res)) +
      geom_point(color='light blue',alpha=0.7) +
      geom_smooth(method="lm")+
      theme_light()
    plot(residPlot)

    residHistogram = ggplot(dataPlot,aes(x=res)) +
      geom_histogram(aes(y=..density..),fill='light blue',alpha=1) +
      geom_density(color='lightblue4') +
      theme_light()
    plot(residHistogram)
    
    print("Coefficients") 
    #no interval for glmnet: https://stackoverflow.com/questions/39750965/confidence-intervals-for-ridge-regression
    t=coef(model,s=model.caret$bestTune$lambda)
    model.coef = t[which(t[,1]!=0),]
    print(as.data.frame(model.coef))
    id = NULL # not really needed but added for consistency
    return(list(model = model.caret,id = id, residPlot = residPlot, metricsPlot=metricsPlot ))
  }
  if (method == 'lars'){
    print(model.caret)
    print(plot(model.caret))
    print(model.caret$bestTune)
    
    # Metrics Plot
    dataPlot = model.caret$results %>%
        gather(key='metric',value='value',-fraction) %>%
      dplyr::filter(metric %in% c('MAE','RMSE','Rsquared'))
    metricsPlot = ggplot(data=dataPlot,aes(x=fraction,y=value) ) +
      geom_line(color='lightblue4') +
      geom_point(color='blue',alpha=0.7,size=.9) +
      facet_wrap(~metric,ncol=4,scales='free_y')+
      theme_light()
    plot(metricsPlot)
    
    # Residuals Plot
    dataPlot=data.frame(pred=predict(model.caret,data),res=resid(model.caret))
    residPlot = ggplot(dataPlot,aes(x=pred,y=res)) +
      geom_point(color='light blue',alpha=0.7) +
      geom_smooth(method="lm")+
      theme_light()
    plot(residPlot)

    residHistogram = ggplot(dataPlot,aes(x=res)) +
      geom_histogram(aes(y=..density..),fill='light blue',alpha=1) +
      geom_density(color='lightblue4') + 
      theme_light()
    plot(residHistogram)
    
    print("Coefficients") 
    t=coef(model.caret$finalModel,s=model.caret$bestTune$fraction,mode='fraction')
    model.coef = t[which(t!=0)]
    print(model.coef)
    id = NULL # not really needed but added for consistency
    return(list(model = model.caret,id = id, residPlot = residPlot, residHistogram=residHistogram))
  }
}

# https://stackoverflow.com/questions/48265743/linear-model-subset-selection-goodness-of-fit-with-k-fold-cross-validation
# changed slightly since call[[2]] was just returning "formula" without actually returnign the value in formula
predict.regsubsets <- function(object, newdata, id, formula, ...) {
    #form <- as.formula(object$call[[2]])
    mat <- model.matrix(formula, newdata) # adds intercept and expands any interaction terms
    coefi <- coef(object, id = id)
    xvars <- names(coefi)
    return(mat[,xvars]%*%coefi)
}
  
test.model = function(model, test, level=0.95
                      ,draw.limits = FALSE, good = 0.1, ok = 0.15
                      ,method = NULL, subopt = NULL
                      ,id = NULL, formula, feature.names, label.names
                      ,transformation = NULL){
  ## if using caret for glm select equivalent functionality, 
  ## need to pass formula (full is ok as it will select subset of variables from there)
  if (is.null(method)){
    pred = predict(model, newdata=test, interval="confidence", level = level) 
  }
  
  if (method == 'leapForward' | method == 'leapBackward' | method == 'leapSeq'){
    pred = predict.regsubsets(model, newdata = test, id = id, formula = formula)
  }
  
  if (method == 'glmnet' && subopt == 'LASSO'){
    xtest = as.matrix(test[,feature.names]) 
    pred=as.data.frame(predict(model, xtest))
  }
  
  if (method == 'lars'){
    pred=as.data.frame(predict(model, newdata = test))
  }
    
  # Summary of predicted values
  print ("Summary of predicted values: ")
  print(summary(pred[,1]))

  test.mse = mean((test[,label.names]-pred[,1])^2)
  print (paste(method, subopt, "Test MSE:", test.mse, sep=" "))
  
  if(log.pred == TRUE || norm.pred == TRUE){
    # plot transformewd comparison first
    plot(test[,label.names],pred[,1],xlab = "Actual (Transformed)", ylab = "Predicted (Transformed)")
    abline(coef=c(0,1),col='blue')
  }
    
  if (log.pred == FALSE && norm.pred == FALSE){
    x = test[,label.names]
    y = pred[,1]
  }
  if (log.pred == TRUE){
    x = 10^test[,label.names]
    y = 10^pred[,1]  
  }
  if (norm.pred == TRUE){
    x = predict(transformation, test[,label.names], inverse = TRUE)
    y = predict(transformation, pred[,1], inverse = TRUE)
  }

  plot(x, y, xlab = "Actual", ylab = "Predicted")
  abline(0,(1+good),col='green', lwd = 3)
  abline(0,(1-good),col='green', lwd = 3)
  abline(0,(1+ok),col='blue', lwd = 3)
  abline(0,(1-ok),col='blue', lwd = 3)
  
}

Setup Formulae

n <- names(data.train)
 formula <- as.formula(paste(paste(n[n %in% label.names], collapse = " + ")
                             ," ~", paste(n[!n %in% label.names], collapse = " + "))) 

grand.mean.formula = as.formula(paste(paste(n[n %in% label.names], collapse = " + ")," ~ 1"))

print(formula)

## y3.log ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + 
##     x12 + x13 + x14 + x15 + x16 + x17 + x19 + x20 + x21 + x22 + 
##     x23 + stat1 + stat2 + stat3 + stat4 + stat5 + stat6 + stat7 + 
##     stat8 + stat9 + stat10 + stat11 + stat12 + stat13 + stat14 + 
##     stat15 + stat16 + stat17 + stat18 + stat19 + stat20 + stat21 + 
##     stat22 + stat23 + stat24 + stat25 + stat26 + stat27 + stat28 + 
##     stat29 + stat30 + stat31 + stat32 + stat33 + stat34 + stat35 + 
##     stat36 + stat37 + stat38 + stat39 + stat40 + stat41 + stat42 + 
##     stat43 + stat44 + stat45 + stat46 + stat47 + stat48 + stat49 + 
##     stat50 + stat51 + stat52 + stat53 + stat54 + stat55 + stat56 + 
##     stat57 + stat58 + stat59 + stat60 + stat61 + stat62 + stat63 + 
##     stat64 + stat65 + stat66 + stat67 + stat68 + stat69 + stat70 + 
##     stat71 + stat72 + stat73 + stat74 + stat75 + stat76 + stat77 + 
##     stat78 + stat79 + stat80 + stat81 + stat82 + stat83 + stat84 + 
##     stat85 + stat86 + stat87 + stat88 + stat89 + stat90 + stat91 + 
##     stat92 + stat93 + stat94 + stat95 + stat96 + stat97 + stat98 + 
##     stat99 + stat100 + stat101 + stat102 + stat103 + stat104 + 
##     stat105 + stat106 + stat107 + stat108 + stat109 + stat110 + 
##     stat111 + stat112 + stat113 + stat114 + stat115 + stat116 + 
##     stat117 + stat118 + stat119 + stat120 + stat121 + stat122 + 
##     stat123 + stat124 + stat125 + stat126 + stat127 + stat128 + 
##     stat129 + stat130 + stat131 + stat132 + stat133 + stat134 + 
##     stat135 + stat136 + stat137 + stat138 + stat139 + stat140 + 
##     stat141 + stat142 + stat143 + stat144 + stat145 + stat146 + 
##     stat147 + stat148 + stat149 + stat150 + stat151 + stat152 + 
##     stat153 + stat154 + stat155 + stat156 + stat157 + stat158 + 
##     stat159 + stat160 + stat161 + stat162 + stat163 + stat164 + 
##     stat165 + stat166 + stat167 + stat168 + stat169 + stat170 + 
##     stat171 + stat172 + stat173 + stat174 + stat175 + stat176 + 
##     stat177 + stat178 + stat179 + stat180 + stat181 + stat182 + 
##     stat183 + stat184 + stat185 + stat186 + stat187 + stat188 + 
##     stat189 + stat190 + stat191 + stat192 + stat193 + stat194 + 
##     stat195 + stat196 + stat197 + stat198 + stat199 + stat200 + 
##     stat201 + stat202 + stat203 + stat204 + stat205 + stat206 + 
##     stat207 + stat208 + stat209 + stat210 + stat211 + stat212 + 
##     stat213 + stat214 + stat215 + stat216 + stat217 + x18.sqrt

print(grand.mean.formula)

## y3.log ~ 1

# Update feature.names because we may have transformed some features
feature.names = n[!n %in% label.names]

Full Model

model.full = lm(formula , data.train)
summary(model.full)

## 
## Call:
## lm(formula = formula, data = data.train)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.084270 -0.020887 -0.004679  0.016324  0.192421 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.964e+00  9.567e-03 205.264  < 2e-16 ***
## x1           3.753e-05  6.551e-04   0.057 0.954317    
## x2           2.546e-04  4.200e-04   0.606 0.544424    
## x3          -4.489e-05  1.151e-04  -0.390 0.696427    
## x4          -5.191e-05  9.107e-06  -5.700 1.26e-08 ***
## x5           4.627e-05  2.971e-04   0.156 0.876224    
## x6           2.467e-05  5.966e-04   0.041 0.967015    
## x7           1.160e-02  6.401e-04  18.126  < 2e-16 ***
## x8           3.561e-04  1.496e-04   2.380 0.017356 *  
## x9           3.273e-03  3.327e-04   9.837  < 2e-16 ***
## x10          1.439e-03  3.093e-04   4.653 3.35e-06 ***
## x11          2.285e+05  7.408e+04   3.085 0.002047 ** 
## x12         -3.337e-05  1.891e-04  -0.176 0.859989    
## x13          5.557e-05  7.523e-05   0.739 0.460115    
## x14         -4.703e-04  3.247e-04  -1.448 0.147558    
## x15          8.258e-05  3.104e-04   0.266 0.790195    
## x16          9.426e-04  2.144e-04   4.396 1.12e-05 ***
## x17          1.667e-03  3.281e-04   5.080 3.90e-07 ***
## x19          3.192e-04  1.668e-04   1.913 0.055799 .  
## x20         -2.265e-04  1.151e-03  -0.197 0.844078    
## x21          1.585e-04  4.231e-05   3.746 0.000182 ***
## x22         -1.857e-04  3.464e-04  -0.536 0.591929    
## x23         -1.755e-04  3.302e-04  -0.531 0.595166    
## stat1       -8.102e-05  2.513e-04  -0.322 0.747132    
## stat2        1.793e-04  2.477e-04   0.724 0.469244    
## stat3        3.246e-04  2.502e-04   1.297 0.194589    
## stat4       -5.361e-04  2.514e-04  -2.132 0.033050 *  
## stat5       -7.079e-05  2.495e-04  -0.284 0.776623    
## stat6       -1.708e-04  2.493e-04  -0.685 0.493322    
## stat7       -3.447e-04  2.495e-04  -1.382 0.167084    
## stat8        4.211e-04  2.491e-04   1.690 0.091004 .  
## stat9       -4.007e-05  2.498e-04  -0.160 0.872552    
## stat10      -3.630e-04  2.500e-04  -1.452 0.146454    
## stat11      -3.163e-04  2.521e-04  -1.255 0.209528    
## stat12       3.063e-04  2.479e-04   1.236 0.216586    
## stat13      -6.601e-04  2.481e-04  -2.660 0.007830 ** 
## stat14      -1.024e-03  2.478e-04  -4.134 3.61e-05 ***
## stat15      -2.154e-04  2.478e-04  -0.869 0.384789    
## stat16       1.297e-04  2.498e-04   0.519 0.603682    
## stat17      -7.625e-05  2.467e-04  -0.309 0.757256    
## stat18      -4.193e-04  2.490e-04  -1.684 0.092201 .  
## stat19       1.981e-04  2.483e-04   0.798 0.425017    
## stat20      -2.986e-04  2.491e-04  -1.199 0.230564    
## stat21      -1.034e-04  2.500e-04  -0.414 0.679204    
## stat22      -4.834e-04  2.496e-04  -1.937 0.052823 .  
## stat23       4.457e-04  2.489e-04   1.791 0.073401 .  
## stat24      -4.723e-04  2.512e-04  -1.881 0.060068 .  
## stat25      -3.752e-04  2.488e-04  -1.508 0.131528    
## stat26      -2.645e-04  2.490e-04  -1.063 0.288017    
## stat27       3.282e-04  2.486e-04   1.321 0.186709    
## stat28       4.716e-05  2.486e-04   0.190 0.849529    
## stat29       2.858e-04  2.513e-04   1.137 0.255497    
## stat30       3.608e-04  2.528e-04   1.427 0.153617    
## stat31       9.486e-05  2.514e-04   0.377 0.705976    
## stat32      -2.828e-05  2.514e-04  -0.113 0.910430    
## stat33      -2.987e-04  2.483e-04  -1.203 0.229105    
## stat34      -4.335e-05  2.494e-04  -0.174 0.862017    
## stat35      -2.733e-04  2.500e-04  -1.093 0.274286    
## stat36       1.840e-04  2.470e-04   0.745 0.456316    
## stat37      -2.395e-04  2.502e-04  -0.957 0.338598    
## stat38       8.622e-05  2.503e-04   0.344 0.730545    
## stat39      -1.508e-04  2.478e-04  -0.608 0.542894    
## stat40       2.898e-04  2.485e-04   1.166 0.243603    
## stat41      -3.423e-04  2.474e-04  -1.384 0.166408    
## stat42       2.284e-04  2.481e-04   0.920 0.357357    
## stat43      -2.561e-04  2.488e-04  -1.029 0.303442    
## stat44       6.146e-05  2.501e-04   0.246 0.805933    
## stat45      -1.661e-04  2.497e-04  -0.665 0.505862    
## stat46       4.545e-04  2.489e-04   1.826 0.067967 .  
## stat47       2.701e-04  2.506e-04   1.078 0.281133    
## stat48       2.719e-04  2.499e-04   1.088 0.276618    
## stat49       2.534e-04  2.478e-04   1.023 0.306560    
## stat50       2.624e-04  2.483e-04   1.057 0.290759    
## stat51       1.792e-04  2.508e-04   0.715 0.474843    
## stat52      -7.772e-05  2.491e-04  -0.312 0.755068    
## stat53      -1.731e-05  2.510e-04  -0.069 0.945000    
## stat54      -2.559e-04  2.522e-04  -1.015 0.310325    
## stat55       1.420e-04  2.473e-04   0.574 0.565822    
## stat56       1.711e-04  2.503e-04   0.683 0.494335    
## stat57       1.389e-04  2.461e-04   0.564 0.572656    
## stat58      -4.820e-05  2.481e-04  -0.194 0.845983    
## stat59       2.231e-04  2.486e-04   0.897 0.369566    
## stat60       5.723e-04  2.502e-04   2.287 0.022217 *  
## stat61      -3.785e-04  2.505e-04  -1.511 0.130817    
## stat62      -7.639e-05  2.496e-04  -0.306 0.759559    
## stat63       1.609e-04  2.496e-04   0.644 0.519345    
## stat64       5.628e-05  2.488e-04   0.226 0.821041    
## stat65      -3.687e-04  2.514e-04  -1.466 0.142647    
## stat66       1.440e-04  2.533e-04   0.568 0.569744    
## stat67       1.391e-05  2.509e-04   0.055 0.955795    
## stat68       8.835e-05  2.491e-04   0.355 0.722802    
## stat69       4.914e-05  2.492e-04   0.197 0.843688    
## stat70       5.980e-05  2.492e-04   0.240 0.810323    
## stat71       6.225e-05  2.491e-04   0.250 0.802634    
## stat72       3.549e-04  2.499e-04   1.420 0.155591    
## stat73       3.397e-04  2.498e-04   1.360 0.173947    
## stat74      -1.966e-04  2.497e-04  -0.787 0.431241    
## stat75      -1.799e-04  2.509e-04  -0.717 0.473340    
## stat76       9.648e-05  2.491e-04   0.387 0.698537    
## stat77      -2.049e-05  2.484e-04  -0.083 0.934250    
## stat78      -1.670e-04  2.506e-04  -0.666 0.505282    
## stat79      -1.250e-04  2.510e-04  -0.498 0.618498    
## stat80       7.305e-06  2.502e-04   0.029 0.976705    
## stat81       3.807e-05  2.507e-04   0.152 0.879315    
## stat82       3.161e-04  2.497e-04   1.266 0.205538    
## stat83      -3.892e-04  2.501e-04  -1.556 0.119666    
## stat84      -1.027e-04  2.486e-04  -0.413 0.679577    
## stat85      -1.224e-04  2.505e-04  -0.489 0.625115    
## stat86      -3.569e-05  2.495e-04  -0.143 0.886272    
## stat87      -2.440e-04  2.502e-04  -0.975 0.329438    
## stat88       4.157e-05  2.474e-04   0.168 0.866550    
## stat89      -1.942e-04  2.484e-04  -0.782 0.434409    
## stat90      -1.349e-04  2.486e-04  -0.543 0.587436    
## stat91      -3.840e-04  2.483e-04  -1.546 0.122138    
## stat92      -4.778e-04  2.503e-04  -1.909 0.056320 .  
## stat93      -1.447e-04  2.531e-04  -0.572 0.567445    
## stat94      -1.588e-04  2.502e-04  -0.635 0.525762    
## stat95      -4.992e-05  2.496e-04  -0.200 0.841480    
## stat96      -5.012e-04  2.484e-04  -2.017 0.043704 *  
## stat97       1.285e-04  2.479e-04   0.518 0.604236    
## stat98       3.677e-03  2.458e-04  14.957  < 2e-16 ***
## stat99       3.027e-04  2.522e-04   1.200 0.230226    
## stat100      7.155e-04  2.505e-04   2.856 0.004308 ** 
## stat101     -2.865e-04  2.526e-04  -1.134 0.256749    
## stat102      1.537e-04  2.501e-04   0.615 0.538858    
## stat103     -2.925e-04  2.516e-04  -1.163 0.245038    
## stat104     -1.912e-04  2.478e-04  -0.772 0.440278    
## stat105      1.064e-04  2.470e-04   0.431 0.666707    
## stat106     -3.019e-04  2.507e-04  -1.204 0.228632    
## stat107     -2.986e-04  2.471e-04  -1.208 0.227032    
## stat108     -2.424e-04  2.492e-04  -0.973 0.330678    
## stat109     -2.995e-04  2.492e-04  -1.201 0.229635    
## stat110     -3.341e-03  2.472e-04 -13.518  < 2e-16 ***
## stat111     -1.863e-04  2.495e-04  -0.747 0.455357    
## stat112      2.893e-05  2.500e-04   0.116 0.907874    
## stat113     -3.143e-04  2.505e-04  -1.255 0.209607    
## stat114      2.477e-04  2.487e-04   0.996 0.319458    
## stat115      1.740e-04  2.496e-04   0.697 0.485900    
## stat116      3.472e-04  2.495e-04   1.391 0.164140    
## stat117     -1.056e-04  2.506e-04  -0.421 0.673604    
## stat118     -3.167e-04  2.465e-04  -1.285 0.199005    
## stat119      2.256e-05  2.482e-04   0.091 0.927581    
## stat120      1.505e-04  2.475e-04   0.608 0.543062    
## stat121     -6.510e-05  2.505e-04  -0.260 0.794990    
## stat122      3.297e-05  2.486e-04   0.133 0.894502    
## stat123      1.158e-04  2.533e-04   0.457 0.647620    
## stat124     -1.758e-04  2.495e-04  -0.704 0.481227    
## stat125      1.325e-04  2.494e-04   0.531 0.595182    
## stat126      1.315e-04  2.489e-04   0.528 0.597194    
## stat127      2.754e-04  2.487e-04   1.107 0.268300    
## stat128     -2.167e-04  2.487e-04  -0.872 0.383479    
## stat129      2.801e-04  2.484e-04   1.128 0.259534    
## stat130      2.470e-04  2.506e-04   0.986 0.324417    
## stat131      3.025e-04  2.495e-04   1.213 0.225291    
## stat132     -1.310e-04  2.476e-04  -0.529 0.596796    
## stat133     -3.458e-05  2.482e-04  -0.139 0.889202    
## stat134     -1.031e-04  2.485e-04  -0.415 0.678221    
## stat135     -5.921e-05  2.487e-04  -0.238 0.811829    
## stat136     -1.962e-05  2.504e-04  -0.078 0.937543    
## stat137     -4.409e-05  2.488e-04  -0.177 0.859316    
## stat138      2.106e-04  2.481e-04   0.849 0.396006    
## stat139      2.304e-04  2.518e-04   0.915 0.360262    
## stat140      1.903e-04  2.485e-04   0.766 0.443779    
## stat141      8.273e-05  2.477e-04   0.334 0.738398    
## stat142     -1.681e-04  2.515e-04  -0.668 0.503886    
## stat143     -1.514e-05  2.486e-04  -0.061 0.951431    
## stat144      1.104e-04  2.486e-04   0.444 0.656869    
## stat145      6.258e-05  2.532e-04   0.247 0.804794    
## stat146     -7.478e-04  2.510e-04  -2.980 0.002900 ** 
## stat147     -4.528e-04  2.508e-04  -1.806 0.071014 .  
## stat148     -5.320e-04  2.479e-04  -2.146 0.031958 *  
## stat149     -4.307e-04  2.508e-04  -1.717 0.085960 .  
## stat150     -2.725e-05  2.502e-04  -0.109 0.913277    
## stat151     -2.918e-05  2.504e-04  -0.117 0.907209    
## stat152     -2.460e-04  2.470e-04  -0.996 0.319277    
## stat153     -1.520e-04  2.536e-04  -0.600 0.548856    
## stat154     -3.719e-05  2.519e-04  -0.148 0.882622    
## stat155      2.738e-04  2.493e-04   1.098 0.272204    
## stat156      4.850e-04  2.506e-04   1.936 0.052945 .  
## stat157     -7.551e-05  2.475e-04  -0.305 0.760355    
## stat158      1.549e-04  2.515e-04   0.616 0.538052    
## stat159     -5.162e-05  2.487e-04  -0.208 0.835556    
## stat160      1.935e-04  2.503e-04   0.773 0.439463    
## stat161      7.585e-05  2.506e-04   0.303 0.762152    
## stat162     -8.382e-05  2.493e-04  -0.336 0.736753    
## stat163      1.656e-04  2.514e-04   0.659 0.510114    
## stat164      2.990e-04  2.525e-04   1.184 0.236425    
## stat165     -5.746e-05  2.464e-04  -0.233 0.815625    
## stat166      8.754e-06  2.465e-04   0.036 0.971672    
## stat167     -5.479e-05  2.499e-04  -0.219 0.826503    
## stat168     -3.103e-04  2.478e-04  -1.252 0.210623    
## stat169     -1.230e-04  2.490e-04  -0.494 0.621413    
## stat170     -1.097e-04  2.498e-04  -0.439 0.660663    
## stat171      1.451e-04  2.512e-04   0.578 0.563474    
## stat172      3.834e-04  2.478e-04   1.547 0.121903    
## stat173     -1.797e-04  2.503e-04  -0.718 0.472868    
## stat174      1.462e-04  2.496e-04   0.586 0.558146    
## stat175     -3.910e-04  2.513e-04  -1.556 0.119774    
## stat176      1.150e-04  2.502e-04   0.460 0.645865    
## stat177     -1.341e-04  2.503e-04  -0.536 0.592190    
## stat178     -2.490e-04  2.542e-04  -0.980 0.327359    
## stat179     -2.147e-05  2.482e-04  -0.086 0.931073    
## stat180     -1.797e-04  2.475e-04  -0.726 0.467864    
## stat181      2.657e-04  2.511e-04   1.059 0.289866    
## stat182      3.110e-04  2.523e-04   1.233 0.217713    
## stat183      7.981e-05  2.470e-04   0.323 0.746629    
## stat184      1.852e-04  2.501e-04   0.741 0.458990    
## stat185     -1.287e-04  2.468e-04  -0.522 0.601945    
## stat186     -1.686e-04  2.515e-04  -0.670 0.502683    
## stat187     -3.241e-04  2.480e-04  -1.307 0.191298    
## stat188     -1.924e-04  2.477e-04  -0.777 0.437341    
## stat189      1.759e-04  2.501e-04   0.703 0.481810    
## stat190      2.818e-04  2.483e-04   1.135 0.256428    
## stat191     -2.591e-04  2.485e-04  -1.043 0.297165    
## stat192      5.169e-06  2.512e-04   0.021 0.983584    
## stat193      2.998e-05  2.524e-04   0.119 0.905447    
## stat194      7.698e-05  2.483e-04   0.310 0.756532    
## stat195      4.011e-04  2.501e-04   1.604 0.108844    
## stat196     -1.110e-04  2.534e-04  -0.438 0.661467    
## stat197     -1.750e-05  2.458e-04  -0.071 0.943264    
## stat198     -1.833e-04  2.506e-04  -0.731 0.464545    
## stat199      2.582e-04  2.476e-04   1.043 0.296995    
## stat200     -2.173e-04  2.471e-04  -0.880 0.379152    
## stat201      3.201e-05  2.486e-04   0.129 0.897559    
## stat202     -4.217e-04  2.523e-04  -1.671 0.094732 .  
## stat203      3.127e-05  2.481e-04   0.126 0.899708    
## stat204     -3.796e-04  2.472e-04  -1.535 0.124777    
## stat205      5.813e-05  2.485e-04   0.234 0.815081    
## stat206     -8.203e-05  2.518e-04  -0.326 0.744628    
## stat207      3.055e-04  2.474e-04   1.235 0.217043    
## stat208      1.132e-04  2.499e-04   0.453 0.650742    
## stat209     -1.474e-04  2.487e-04  -0.593 0.553526    
## stat210     -1.837e-04  2.493e-04  -0.737 0.461180    
## stat211     -7.206e-05  2.488e-04  -0.290 0.772095    
## stat212     -2.248e-05  2.474e-04  -0.091 0.927616    
## stat213      8.886e-05  2.500e-04   0.355 0.722307    
## stat214     -3.070e-04  2.498e-04  -1.229 0.219152    
## stat215     -1.531e-04  2.505e-04  -0.611 0.540925    
## stat216     -1.137e-04  2.500e-04  -0.455 0.649258    
## stat217      3.294e-04  2.504e-04   1.315 0.188437    
## x18.sqrt     2.653e-02  9.464e-04  28.032  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03158 on 5343 degrees of freedom
## Multiple R-squared:  0.2757, Adjusted R-squared:  0.2432 
## F-statistic: 8.476 on 240 and 5343 DF,  p-value: < 2.2e-16

cd.full = plot.diagnostics(model.full, data.train)

## [1] "Number of data points that have Cook's D > 4/n: 284"
## [1] "Number of data points that have Cook's D > 1: 0"

Checking with removal of high influence points

high.cd = names(cd.full[cd.full > 4/nrow(data.train)])
data.train2 = data.train[!(rownames(data.train)) %in% high.cd,]
model.full2 = lm(formula , data.train2)
summary(model.full2)

## 
## Call:
## lm(formula = formula, data = data.train2)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.058186 -0.017664 -0.002528  0.016290  0.069788 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.948e+00  7.867e-03 247.625  < 2e-16 ***
## x1          -6.377e-05  5.386e-04  -0.118 0.905767    
## x2           2.759e-04  3.445e-04   0.801 0.423264    
## x3          -1.922e-05  9.402e-05  -0.204 0.838044    
## x4          -5.981e-05  7.502e-06  -7.972 1.91e-15 ***
## x5           4.093e-04  2.436e-04   1.680 0.092950 .  
## x6          -3.784e-04  4.896e-04  -0.773 0.439593    
## x7           1.217e-02  5.249e-04  23.190  < 2e-16 ***
## x8           4.293e-04  1.230e-04   3.491 0.000485 ***
## x9           3.296e-03  2.725e-04  12.093  < 2e-16 ***
## x10          1.873e-03  2.542e-04   7.369 2.00e-13 ***
## x11          2.875e+05  6.094e+04   4.717 2.45e-06 ***
## x12          5.809e-05  1.547e-04   0.376 0.707213    
## x13          8.143e-05  6.183e-05   1.317 0.187927    
## x14         -3.772e-04  2.659e-04  -1.419 0.156013    
## x15          1.344e-04  2.546e-04   0.528 0.597683    
## x16          1.075e-03  1.760e-04   6.108 1.09e-09 ***
## x17          1.762e-03  2.694e-04   6.541 6.71e-11 ***
## x19          2.156e-04  1.371e-04   1.573 0.115750    
## x20         -2.731e-04  9.462e-04  -0.289 0.772870    
## x21          1.471e-04  3.477e-05   4.230 2.38e-05 ***
## x22         -3.615e-04  2.837e-04  -1.274 0.202713    
## x23         -2.388e-05  2.716e-04  -0.088 0.929919    
## stat1       -2.044e-04  2.060e-04  -0.992 0.321090    
## stat2        1.556e-04  2.030e-04   0.767 0.443366    
## stat3        3.794e-04  2.052e-04   1.849 0.064496 .  
## stat4       -6.022e-04  2.072e-04  -2.906 0.003671 ** 
## stat5       -2.007e-04  2.051e-04  -0.979 0.327713    
## stat6       -1.305e-04  2.045e-04  -0.638 0.523420    
## stat7       -3.877e-04  2.044e-04  -1.896 0.057964 .  
## stat8        2.705e-04  2.043e-04   1.324 0.185451    
## stat9       -1.525e-05  2.056e-04  -0.074 0.940859    
## stat10      -2.589e-04  2.050e-04  -1.263 0.206600    
## stat11      -4.472e-04  2.073e-04  -2.157 0.031055 *  
## stat12       1.503e-04  2.036e-04   0.738 0.460322    
## stat13      -6.610e-04  2.035e-04  -3.248 0.001168 ** 
## stat14      -9.847e-04  2.032e-04  -4.846 1.30e-06 ***
## stat15      -4.366e-04  2.037e-04  -2.143 0.032129 *  
## stat16      -5.448e-05  2.048e-04  -0.266 0.790233    
## stat17      -2.621e-04  2.028e-04  -1.292 0.196352    
## stat18      -3.863e-04  2.043e-04  -1.891 0.058621 .  
## stat19       2.872e-04  2.043e-04   1.405 0.159974    
## stat20       1.593e-04  2.046e-04   0.779 0.436239    
## stat21      -1.963e-04  2.054e-04  -0.956 0.339269    
## stat22      -3.181e-04  2.048e-04  -1.553 0.120506    
## stat23       4.623e-04  2.046e-04   2.259 0.023911 *  
## stat24      -5.004e-04  2.064e-04  -2.424 0.015374 *  
## stat25      -3.786e-04  2.040e-04  -1.856 0.063458 .  
## stat26      -3.661e-04  2.048e-04  -1.788 0.073911 .  
## stat27       2.035e-04  2.044e-04   0.996 0.319532    
## stat28      -4.105e-05  2.045e-04  -0.201 0.840897    
## stat29       3.222e-04  2.063e-04   1.562 0.118341    
## stat30       2.311e-04  2.069e-04   1.117 0.264141    
## stat31       1.102e-04  2.065e-04   0.534 0.593483    
## stat32      -7.713e-06  2.065e-04  -0.037 0.970203    
## stat33      -3.347e-04  2.042e-04  -1.639 0.101201    
## stat34       2.860e-04  2.052e-04   1.394 0.163456    
## stat35      -2.654e-04  2.055e-04  -1.291 0.196664    
## stat36       4.764e-05  2.031e-04   0.235 0.814537    
## stat37      -5.217e-05  2.058e-04  -0.254 0.799879    
## stat38       2.283e-04  2.054e-04   1.112 0.266371    
## stat39      -2.326e-04  2.030e-04  -1.146 0.251821    
## stat40       1.871e-04  2.043e-04   0.916 0.359863    
## stat41      -4.275e-04  2.028e-04  -2.108 0.035118 *  
## stat42       2.533e-04  2.039e-04   1.242 0.214211    
## stat43      -3.345e-04  2.043e-04  -1.637 0.101623    
## stat44       1.873e-04  2.056e-04   0.911 0.362389    
## stat45       3.781e-05  2.051e-04   0.184 0.853753    
## stat46       3.805e-04  2.045e-04   1.861 0.062842 .  
## stat47       2.096e-04  2.057e-04   1.019 0.308342    
## stat48       3.035e-04  2.046e-04   1.483 0.138131    
## stat49       6.491e-05  2.038e-04   0.319 0.750061    
## stat50       3.134e-04  2.041e-04   1.535 0.124814    
## stat51       4.548e-05  2.060e-04   0.221 0.825298    
## stat52       1.376e-07  2.047e-04   0.001 0.999464    
## stat53      -7.636e-05  2.061e-04  -0.371 0.710990    
## stat54      -3.635e-04  2.077e-04  -1.750 0.080127 .  
## stat55       1.442e-04  2.032e-04   0.709 0.478063    
## stat56       2.929e-04  2.056e-04   1.425 0.154307    
## stat57       9.230e-05  2.026e-04   0.456 0.648714    
## stat58      -7.732e-05  2.035e-04  -0.380 0.704048    
## stat59       2.015e-04  2.043e-04   0.986 0.323978    
## stat60       6.398e-04  2.054e-04   3.116 0.001845 ** 
## stat61      -4.421e-04  2.056e-04  -2.150 0.031617 *  
## stat62      -2.611e-04  2.046e-04  -1.276 0.201885    
## stat63       5.829e-05  2.052e-04   0.284 0.776353    
## stat64       2.394e-04  2.044e-04   1.171 0.241500    
## stat65      -2.207e-04  2.064e-04  -1.069 0.284978    
## stat66       1.747e-04  2.082e-04   0.839 0.401419    
## stat67       1.610e-04  2.059e-04   0.782 0.434329    
## stat68      -1.423e-06  2.047e-04  -0.007 0.994453    
## stat69      -1.192e-04  2.046e-04  -0.583 0.560168    
## stat70       1.605e-04  2.049e-04   0.783 0.433509    
## stat71       2.712e-05  2.050e-04   0.132 0.894755    
## stat72       1.781e-04  2.052e-04   0.868 0.385566    
## stat73       3.713e-04  2.054e-04   1.808 0.070631 .  
## stat74      -7.966e-05  2.049e-04  -0.389 0.697502    
## stat75       3.440e-05  2.062e-04   0.167 0.867507    
## stat76       1.330e-04  2.045e-04   0.650 0.515611    
## stat77       2.264e-04  2.041e-04   1.109 0.267370    
## stat78      -4.841e-04  2.054e-04  -2.357 0.018439 *  
## stat79       6.126e-05  2.057e-04   0.298 0.765840    
## stat80      -6.550e-05  2.056e-04  -0.319 0.750049    
## stat81      -4.088e-05  2.065e-04  -0.198 0.843084    
## stat82       1.639e-04  2.051e-04   0.799 0.424292    
## stat83      -4.428e-04  2.055e-04  -2.155 0.031206 *  
## stat84      -2.882e-04  2.041e-04  -1.412 0.157941    
## stat85      -3.378e-04  2.059e-04  -1.641 0.100952    
## stat86       1.547e-04  2.048e-04   0.755 0.450295    
## stat87      -2.072e-04  2.057e-04  -1.007 0.313799    
## stat88       1.661e-04  2.032e-04   0.817 0.413950    
## stat89       8.700e-05  2.047e-04   0.425 0.670831    
## stat90      -2.455e-04  2.045e-04  -1.201 0.229951    
## stat91      -3.149e-04  2.037e-04  -1.546 0.122057    
## stat92      -4.849e-04  2.054e-04  -2.361 0.018260 *  
## stat93      -1.245e-04  2.086e-04  -0.597 0.550777    
## stat94       3.791e-05  2.051e-04   0.185 0.853327    
## stat95       5.988e-05  2.055e-04   0.291 0.770809    
## stat96      -4.818e-04  2.041e-04  -2.360 0.018298 *  
## stat97       2.990e-04  2.034e-04   1.470 0.141618    
## stat98       3.625e-03  2.022e-04  17.928  < 2e-16 ***
## stat99       4.306e-04  2.069e-04   2.081 0.037502 *  
## stat100      6.788e-04  2.058e-04   3.299 0.000978 ***
## stat101     -2.650e-04  2.073e-04  -1.278 0.201146    
## stat102      1.553e-04  2.054e-04   0.756 0.449769    
## stat103     -3.842e-04  2.063e-04  -1.863 0.062575 .  
## stat104     -1.700e-04  2.039e-04  -0.834 0.404424    
## stat105      2.073e-04  2.029e-04   1.022 0.306934    
## stat106     -2.872e-04  2.055e-04  -1.397 0.162325    
## stat107     -1.469e-04  2.025e-04  -0.725 0.468368    
## stat108     -5.871e-05  2.048e-04  -0.287 0.774329    
## stat109     -3.722e-04  2.045e-04  -1.820 0.068854 .  
## stat110     -3.385e-03  2.029e-04 -16.682  < 2e-16 ***
## stat111     -1.191e-04  2.045e-04  -0.583 0.560253    
## stat112      9.175e-05  2.057e-04   0.446 0.655645    
## stat113     -2.499e-04  2.055e-04  -1.216 0.223938    
## stat114      2.395e-04  2.045e-04   1.171 0.241579    
## stat115      3.186e-04  2.052e-04   1.552 0.120624    
## stat116      2.966e-04  2.048e-04   1.448 0.147615    
## stat117     -8.793e-05  2.054e-04  -0.428 0.668658    
## stat118     -1.121e-05  2.020e-04  -0.056 0.955733    
## stat119      1.404e-04  2.036e-04   0.689 0.490562    
## stat120      1.780e-05  2.032e-04   0.088 0.930202    
## stat121     -6.201e-06  2.056e-04  -0.030 0.975941    
## stat122     -1.454e-04  2.045e-04  -0.711 0.477177    
## stat123      1.647e-04  2.080e-04   0.792 0.428442    
## stat124     -1.480e-04  2.047e-04  -0.723 0.469716    
## stat125      3.774e-05  2.047e-04   0.184 0.853721    
## stat126      1.502e-04  2.042e-04   0.735 0.462169    
## stat127      2.360e-04  2.041e-04   1.156 0.247626    
## stat128     -5.200e-04  2.041e-04  -2.548 0.010856 *  
## stat129      3.735e-04  2.036e-04   1.834 0.066669 .  
## stat130      1.598e-04  2.056e-04   0.777 0.436996    
## stat131      1.644e-04  2.045e-04   0.804 0.421432    
## stat132     -4.304e-05  2.031e-04  -0.212 0.832172    
## stat133      6.853e-05  2.038e-04   0.336 0.736731    
## stat134     -8.923e-05  2.040e-04  -0.437 0.661898    
## stat135     -1.601e-05  2.042e-04  -0.078 0.937534    
## stat136     -5.440e-05  2.052e-04  -0.265 0.790972    
## stat137      6.551e-05  2.040e-04   0.321 0.748133    
## stat138      1.764e-04  2.038e-04   0.866 0.386648    
## stat139      1.206e-04  2.067e-04   0.584 0.559556    
## stat140      2.511e-04  2.035e-04   1.234 0.217278    
## stat141      1.334e-04  2.032e-04   0.657 0.511472    
## stat142     -1.735e-04  2.067e-04  -0.840 0.401173    
## stat143     -1.753e-04  2.043e-04  -0.858 0.390871    
## stat144      2.109e-04  2.042e-04   1.033 0.301860    
## stat145     -5.852e-05  2.080e-04  -0.281 0.778439    
## stat146     -9.620e-04  2.058e-04  -4.675 3.02e-06 ***
## stat147     -3.941e-04  2.063e-04  -1.910 0.056135 .  
## stat148     -4.098e-04  2.034e-04  -2.015 0.043982 *  
## stat149     -5.657e-04  2.066e-04  -2.739 0.006194 ** 
## stat150     -2.295e-04  2.058e-04  -1.115 0.264847    
## stat151      2.556e-04  2.058e-04   1.242 0.214391    
## stat152     -2.793e-04  2.026e-04  -1.378 0.168180    
## stat153      1.067e-04  2.080e-04   0.513 0.608117    
## stat154      2.531e-04  2.073e-04   1.221 0.222200    
## stat155      4.028e-04  2.050e-04   1.965 0.049458 *  
## stat156      4.113e-04  2.054e-04   2.002 0.045331 *  
## stat157     -1.189e-04  2.030e-04  -0.586 0.557862    
## stat158      2.777e-04  2.060e-04   1.348 0.177789    
## stat159      2.332e-05  2.042e-04   0.114 0.909103    
## stat160      1.066e-04  2.061e-04   0.517 0.605134    
## stat161      2.413e-05  2.055e-04   0.117 0.906504    
## stat162     -1.212e-04  2.041e-04  -0.594 0.552496    
## stat163      2.225e-04  2.072e-04   1.074 0.282881    
## stat164      5.406e-05  2.079e-04   0.260 0.794844    
## stat165      1.067e-04  2.024e-04   0.527 0.598027    
## stat166      1.204e-04  2.019e-04   0.597 0.550819    
## stat167     -1.307e-04  2.050e-04  -0.638 0.523817    
## stat168     -2.572e-04  2.032e-04  -1.266 0.205654    
## stat169     -3.618e-05  2.051e-04  -0.176 0.859936    
## stat170      3.935e-05  2.054e-04   0.192 0.848078    
## stat171     -3.488e-05  2.062e-04  -0.169 0.865651    
## stat172      6.266e-04  2.029e-04   3.089 0.002019 ** 
## stat173      1.352e-04  2.054e-04   0.658 0.510507    
## stat174      2.411e-04  2.050e-04   1.176 0.239609    
## stat175     -2.703e-04  2.063e-04  -1.310 0.190222    
## stat176     -1.903e-04  2.054e-04  -0.926 0.354270    
## stat177     -4.077e-04  2.054e-04  -1.985 0.047183 *  
## stat178     -9.255e-06  2.085e-04  -0.044 0.964596    
## stat179     -4.820e-05  2.038e-04  -0.236 0.813097    
## stat180     -1.586e-04  2.035e-04  -0.779 0.435778    
## stat181      3.599e-04  2.060e-04   1.747 0.080629 .  
## stat182      4.208e-04  2.076e-04   2.027 0.042747 *  
## stat183      1.673e-04  2.029e-04   0.824 0.409724    
## stat184      2.457e-04  2.051e-04   1.198 0.231005    
## stat185      4.648e-05  2.027e-04   0.229 0.818610    
## stat186      1.194e-04  2.063e-04   0.579 0.562844    
## stat187     -2.259e-04  2.035e-04  -1.110 0.266846    
## stat188     -4.795e-06  2.037e-04  -0.024 0.981220    
## stat189      2.120e-05  2.056e-04   0.103 0.917869    
## stat190      1.154e-04  2.041e-04   0.566 0.571736    
## stat191     -2.161e-04  2.038e-04  -1.060 0.289178    
## stat192     -1.255e-05  2.067e-04  -0.061 0.951563    
## stat193      2.626e-04  2.077e-04   1.264 0.206210    
## stat194      1.943e-06  2.043e-04   0.010 0.992413    
## stat195      8.818e-05  2.056e-04   0.429 0.667997    
## stat196     -1.649e-04  2.076e-04  -0.795 0.426918    
## stat197     -2.453e-04  2.021e-04  -1.214 0.224808    
## stat198     -1.790e-04  2.058e-04  -0.870 0.384479    
## stat199      2.439e-04  2.032e-04   1.200 0.230220    
## stat200     -2.016e-04  2.033e-04  -0.992 0.321360    
## stat201      1.959e-04  2.043e-04   0.959 0.337535    
## stat202     -3.265e-04  2.074e-04  -1.574 0.115472    
## stat203      1.647e-04  2.040e-04   0.807 0.419565    
## stat204     -5.445e-05  2.029e-04  -0.268 0.788489    
## stat205      2.141e-04  2.033e-04   1.053 0.292164    
## stat206     -1.487e-04  2.068e-04  -0.719 0.472283    
## stat207      3.776e-04  2.034e-04   1.856 0.063478 .  
## stat208      4.002e-05  2.053e-04   0.195 0.845481    
## stat209     -7.536e-06  2.038e-04  -0.037 0.970508    
## stat210     -2.480e-04  2.045e-04  -1.213 0.225305    
## stat211     -2.796e-05  2.043e-04  -0.137 0.891175    
## stat212      1.317e-04  2.035e-04   0.647 0.517588    
## stat213      3.248e-05  2.050e-04   0.158 0.874086    
## stat214     -1.896e-04  2.056e-04  -0.922 0.356475    
## stat215     -2.584e-04  2.060e-04  -1.254 0.209832    
## stat216     -1.863e-04  2.046e-04  -0.911 0.362536    
## stat217      1.544e-04  2.052e-04   0.752 0.451934    
## x18.sqrt     2.662e-02  7.742e-04  34.388  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02523 on 5059 degrees of freedom
## Multiple R-squared:  0.3871, Adjusted R-squared:  0.358 
## F-statistic: 13.31 on 240 and 5059 DF,  p-value: < 2.2e-16

cd.full2 = plot.diagnostics(model.full2, data.train2)

## [1] "Number of data points that have Cook's D > 4/n: 291"
## [1] "Number of data points that have Cook's D > 1: 0"

# much more normal residuals than before. 
# Checking to see if distributions are different and if so whcih variables
# High Leverage Plot 
plotData = data.train %>% 
  rownames_to_column() %>%
  mutate(type=ifelse(rowname %in% high.cd,'High','Normal')) %>%
  dplyr::select(type,target=one_of(label.names))

ggplot(data=plotData, aes(x=type,y=target)) +
  geom_boxplot(fill='light blue',outlier.shape=NA) +
  scale_y_continuous(name="Target Variable Values",label=scales::comma_format(accuracy=.1)) +
  theme_light() +
  ggtitle('Distribution of High Leverage Points and Normal  Points')

plotData = data.train %>% 
  rownames_to_column() %>%
  mutate(type=ifelse(rowname %in% high.cd,'High','Normal')) %>%
  dplyr::select(type,one_of(feature.names))

# 2 sample t-tests
comp.test = lapply(dplyr::select(plotData, one_of(feature.names))
                   , function(x) t.test(x ~ plotData$type, var.equal = TRUE)) 

sig.comp = list.filter(comp.test, p.value < 0.05)
sapply(sig.comp, function(x) x[['p.value']])

##           x4           x5          x16       stat15       stat27       stat38       stat98      stat110      stat128 
## 7.538299e-03 3.428941e-02 4.942924e-02 3.782893e-02 3.005070e-02 2.614385e-02 1.238531e-06 4.763874e-03 6.922363e-03 
##      stat146      stat151      stat186      stat193     x18.sqrt 
## 1.415771e-02 3.146729e-02 3.675378e-02 3.756332e-02 1.462902e-02

# Distribution (box) Plots
mm = melt(plotData, id=c('type'))

ggplot(mm) +
  geom_boxplot(aes(x=type, y=value))+
  facet_wrap(~variable, ncol=8, scales = 'free_y') +
  scale_y_continuous(name="values",label=scales::comma_format(accuracy=.1)) +
  ggtitle('Distribution of High Leverage Points and Normal Points')

Grand Means Model

model.null = lm(grand.mean.formula, data.train)
summary(model.null)

## 
## Call:
## lm(formula = grand.mean.formula, data = data.train)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.114816 -0.023952 -0.003362  0.020696  0.190497 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 2.0966918  0.0004858    4316   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0363 on 5583 degrees of freedom

Variable Selection

Basic: http://www.stat.columbia.edu/~martin/W2024/R10.pdf Cross Validation + Other Metrics: http://www.sthda.com/english/articles/37-model-selection-essentials-in-r/154-stepwise-regression-essentials-in-r/

Forward Selection with CV

Train

if (algo.forward.caret == TRUE){
  set.seed(1)
  returned = train.caret.glmselect(formula = formula
                                   , data = data.train
                                   , method = "leapForward"
                                   , feature.names = feature.names)
  model.forward = returned$model
  id = returned$id
}

## Aggregating results
## Selecting tuning parameters
## Fitting nvmax = 15 on full training set
## [1] "All models results"
##     nvmax       RMSE  Rsquared        MAE       RMSESD RsquaredSD        MAESD
## 1       1 0.03426021 0.1102603 0.02678506 0.0007528041 0.02129231 0.0004213658
## 2       2 0.03341255 0.1548136 0.02608465 0.0009549896 0.03477939 0.0005441361
## 3       3 0.03278951 0.1857579 0.02547103 0.0009056890 0.03288428 0.0004986653
## 4       4 0.03227713 0.2113517 0.02470874 0.0008645806 0.03734399 0.0005939377
## 5       5 0.03198524 0.2249611 0.02451264 0.0009072163 0.03436803 0.0005425191
## 6       6 0.03196756 0.2257863 0.02450055 0.0009147022 0.03323145 0.0005522053
## 7       7 0.03186275 0.2306455 0.02443907 0.0008841335 0.03173780 0.0005201270
## 8       8 0.03180485 0.2333632 0.02440347 0.0008909229 0.03190063 0.0004926743
## 9       9 0.03174677 0.2360753 0.02438467 0.0009075872 0.03063354 0.0004965026
## 10     10 0.03168841 0.2387674 0.02432430 0.0008999805 0.02990088 0.0004865291
## 11     11 0.03169907 0.2381655 0.02434296 0.0008523372 0.02756473 0.0004382474
## 12     12 0.03169542 0.2382564 0.02435400 0.0008401842 0.02585450 0.0004245230
## 13     13 0.03167366 0.2392646 0.02432044 0.0008373043 0.02565854 0.0004341691
## 14     14 0.03166420 0.2397724 0.02430440 0.0008508276 0.02621653 0.0004402137
## 15     15 0.03163949 0.2409469 0.02429082 0.0008593315 0.02618453 0.0004442920
## 16     16 0.03165164 0.2403430 0.02429781 0.0008321322 0.02526408 0.0004478658
## 17     17 0.03165851 0.2400597 0.02429486 0.0008510357 0.02571485 0.0004432393
## 18     18 0.03163951 0.2409482 0.02428901 0.0008616926 0.02483458 0.0004450653
## 19     19 0.03167827 0.2390992 0.02432694 0.0008345569 0.02370360 0.0004271269
## 20     20 0.03170627 0.2377880 0.02434710 0.0008290882 0.02341048 0.0004223665
## 21     21 0.03173214 0.2366471 0.02436755 0.0008324223 0.02405134 0.0004327652
## 22     22 0.03176505 0.2350680 0.02439784 0.0008294861 0.02361180 0.0004233768
## 23     23 0.03176223 0.2352378 0.02440165 0.0008317759 0.02380651 0.0004236607
## 24     24 0.03176725 0.2350697 0.02441149 0.0008418167 0.02441508 0.0004374781
## 25     25 0.03178229 0.2344164 0.02442767 0.0008357834 0.02492069 0.0004356016
## 26     26 0.03180622 0.2333191 0.02444664 0.0008366503 0.02485684 0.0004381811
## 27     27 0.03181312 0.2330129 0.02444572 0.0008290087 0.02446424 0.0004249594
## 28     28 0.03182294 0.2326216 0.02443501 0.0008141599 0.02425661 0.0004183447
## 29     29 0.03183304 0.2321204 0.02444154 0.0008082972 0.02396822 0.0004176705
## 30     30 0.03184978 0.2313617 0.02444502 0.0008022351 0.02378389 0.0004065443
## 31     31 0.03185901 0.2309734 0.02444885 0.0007929067 0.02287079 0.0004023088
## 32     32 0.03188304 0.2299186 0.02447488 0.0007895438 0.02342731 0.0004049664
## 33     33 0.03189546 0.2294032 0.02447748 0.0007853423 0.02448265 0.0004046648
## 34     34 0.03190049 0.2291563 0.02447437 0.0007785900 0.02410299 0.0003940673
## 35     35 0.03190137 0.2291546 0.02447921 0.0007786900 0.02397655 0.0003994480
## 36     36 0.03190426 0.2290876 0.02448473 0.0007744877 0.02391424 0.0004010935
## 37     37 0.03190988 0.2289050 0.02447669 0.0007893986 0.02472139 0.0004088587
## 38     38 0.03192742 0.2281280 0.02450082 0.0007844912 0.02473346 0.0004118077
## 39     39 0.03194180 0.2274844 0.02451534 0.0007977205 0.02491037 0.0004218552
## 40     40 0.03195152 0.2270923 0.02452470 0.0008039877 0.02501946 0.0004218392
## 41     41 0.03196520 0.2265038 0.02453214 0.0007948824 0.02484698 0.0004176748
## 42     42 0.03197139 0.2262985 0.02453694 0.0007822986 0.02513960 0.0004144641
## 43     43 0.03198537 0.2256844 0.02454857 0.0007816034 0.02609210 0.0004115544
## 44     44 0.03198423 0.2257095 0.02454886 0.0007914110 0.02553364 0.0004105515
## 45     45 0.03199474 0.2252278 0.02455186 0.0008042751 0.02509173 0.0004165877
## 46     46 0.03200107 0.2249620 0.02455834 0.0008114790 0.02498185 0.0004118621
## 47     47 0.03201505 0.2243482 0.02456905 0.0008170497 0.02417158 0.0004038639
## 48     48 0.03203447 0.2234964 0.02458677 0.0008207037 0.02398047 0.0004118974
## 49     49 0.03204966 0.2227990 0.02459871 0.0008310866 0.02385566 0.0004158852
## 50     50 0.03205584 0.2225217 0.02460689 0.0008276291 0.02425146 0.0004147309
## 51     51 0.03205271 0.2227071 0.02460448 0.0008251882 0.02430544 0.0004031177
## 52     52 0.03205136 0.2228096 0.02460017 0.0008285602 0.02422718 0.0004061043
## 53     53 0.03205729 0.2225258 0.02461368 0.0008369043 0.02348778 0.0003998033
## 54     54 0.03206538 0.2221869 0.02461197 0.0008371078 0.02335276 0.0004055748
## 55     55 0.03207595 0.2217341 0.02461556 0.0008420362 0.02323943 0.0004042657
## 56     56 0.03208132 0.2215685 0.02461177 0.0008305296 0.02288161 0.0004114346
## 57     57 0.03209571 0.2210120 0.02462357 0.0008424689 0.02348665 0.0004368620
## 58     58 0.03210381 0.2206744 0.02463088 0.0008365657 0.02311960 0.0004358985
## 59     59 0.03210164 0.2207993 0.02463965 0.0008315634 0.02308960 0.0004321107
## 60     60 0.03210488 0.2207022 0.02464016 0.0008541732 0.02335477 0.0004559694
## 61     61 0.03210438 0.2207147 0.02463955 0.0008431746 0.02325433 0.0004470284
## 62     62 0.03210947 0.2205243 0.02464388 0.0008414266 0.02302778 0.0004388893
## 63     63 0.03211761 0.2201934 0.02464402 0.0008309068 0.02274551 0.0004248323
## 64     64 0.03212435 0.2199122 0.02464921 0.0008185194 0.02237781 0.0004231796
## 65     65 0.03212728 0.2197865 0.02464404 0.0008220543 0.02252348 0.0004247080
## 66     66 0.03213500 0.2194779 0.02464902 0.0008153190 0.02222005 0.0004030686
## 67     67 0.03213644 0.2194295 0.02464785 0.0008205327 0.02215100 0.0003984378
## 68     68 0.03213748 0.2194193 0.02465613 0.0008242981 0.02228278 0.0004012508
## 69     69 0.03213897 0.2193680 0.02465300 0.0008271265 0.02249565 0.0003992050
## 70     70 0.03214416 0.2191092 0.02465312 0.0008198294 0.02177535 0.0003974562
## 71     71 0.03214006 0.2193066 0.02464383 0.0008278115 0.02207390 0.0004018038
## 72     72 0.03215592 0.2186462 0.02465430 0.0008329419 0.02274761 0.0004085917
## 73     73 0.03216425 0.2183056 0.02465620 0.0008234252 0.02230127 0.0004086639
## 74     74 0.03217120 0.2180078 0.02465456 0.0008355316 0.02192199 0.0004126484
## 75     75 0.03216813 0.2182071 0.02464774 0.0008332710 0.02231807 0.0004173926
## 76     76 0.03217292 0.2180486 0.02465192 0.0008305880 0.02249957 0.0004152097
## 77     77 0.03217278 0.2180923 0.02465874 0.0008292777 0.02273119 0.0004235582
## 78     78 0.03217566 0.2179697 0.02466583 0.0008192179 0.02242350 0.0004253432
## 79     79 0.03219317 0.2172244 0.02467342 0.0008252301 0.02282468 0.0004416766
## 80     80 0.03220749 0.2166414 0.02469160 0.0008181437 0.02265092 0.0004327592
## 81     81 0.03219981 0.2169745 0.02468526 0.0008030665 0.02150155 0.0004127828
## 82     82 0.03219604 0.2171576 0.02467470 0.0007990609 0.02105661 0.0004182189
## 83     83 0.03220285 0.2168978 0.02467794 0.0008034094 0.02131610 0.0004220929
## 84     84 0.03221466 0.2163894 0.02469050 0.0007934029 0.02074806 0.0004122794
## 85     85 0.03220340 0.2168759 0.02468516 0.0007879490 0.02098153 0.0004140607
## 86     86 0.03220948 0.2166152 0.02469995 0.0007892052 0.02120931 0.0004030460
## 87     87 0.03221528 0.2163682 0.02470570 0.0008021915 0.02115657 0.0004113510
## 88     88 0.03222237 0.2160910 0.02471277 0.0008008791 0.02113310 0.0004155982
## 89     89 0.03222398 0.2159829 0.02470692 0.0007999480 0.02086363 0.0004093909
## 90     90 0.03221903 0.2162232 0.02469708 0.0008027121 0.02118674 0.0004170575
## 91     91 0.03222070 0.2161654 0.02469753 0.0008109831 0.02132853 0.0004174175
## 92     92 0.03221782 0.2163085 0.02469331 0.0008159746 0.02128971 0.0004132119
## 93     93 0.03221428 0.2164828 0.02468448 0.0008177147 0.02132864 0.0004152499
## 94     94 0.03221665 0.2163772 0.02469143 0.0008169074 0.02108369 0.0004156402
## 95     95 0.03222475 0.2160430 0.02470089 0.0008209146 0.02097618 0.0004159653
## 96     96 0.03222425 0.2160981 0.02469438 0.0008189753 0.02089167 0.0004144920
## 97     97 0.03222970 0.2158996 0.02470455 0.0008169783 0.02117029 0.0004067262
## 98     98 0.03223003 0.2159077 0.02470973 0.0008166621 0.02142947 0.0004062372
## 99     99 0.03222301 0.2162532 0.02470350 0.0008150071 0.02129735 0.0004006624
## 100   100 0.03223571 0.2157114 0.02471248 0.0008137542 0.02077248 0.0003997836
## 101   101 0.03224534 0.2153242 0.02472065 0.0008091270 0.02086575 0.0003976216
## 102   102 0.03224946 0.2151681 0.02472956 0.0008136732 0.02117533 0.0004047948
## 103   103 0.03224141 0.2155583 0.02471891 0.0008018721 0.02122691 0.0004108949
## 104   104 0.03224212 0.2154790 0.02472284 0.0007962840 0.02099192 0.0004110076
## 105   105 0.03224151 0.2154858 0.02471641 0.0007941398 0.02069080 0.0004136345
## 106   106 0.03224538 0.2153487 0.02471573 0.0007923730 0.02116361 0.0004120177
## 107   107 0.03224810 0.2152266 0.02471626 0.0007838558 0.02086815 0.0004052025
## 108   108 0.03225143 0.2151277 0.02471540 0.0007888712 0.02104432 0.0004138911
## 109   109 0.03225467 0.2149900 0.02471845 0.0007944018 0.02152057 0.0004191947
## 110   110 0.03226019 0.2147780 0.02472409 0.0007967409 0.02145995 0.0004178646
## 111   111 0.03226490 0.2146157 0.02472572 0.0007917044 0.02135171 0.0004251457
## 112   112 0.03226270 0.2147112 0.02472121 0.0007860524 0.02073993 0.0004166495
## 113   113 0.03226559 0.2146022 0.02471867 0.0007916445 0.02068466 0.0004158401
## 114   114 0.03227292 0.2142725 0.02472333 0.0007912540 0.02031648 0.0004084945
## 115   115 0.03227770 0.2140817 0.02472167 0.0007901162 0.02020176 0.0004028207
## 116   116 0.03227340 0.2143063 0.02471510 0.0007937478 0.02055446 0.0004051135
## 117   117 0.03226981 0.2144630 0.02471489 0.0007986030 0.02080442 0.0004072394
## 118   118 0.03226950 0.2144552 0.02471184 0.0007934451 0.02009658 0.0004045565
## 119   119 0.03226662 0.2145847 0.02470615 0.0007892065 0.02000647 0.0003965637
## 120   120 0.03226843 0.2144857 0.02470733 0.0007889785 0.01971223 0.0003900070
## 121   121 0.03228178 0.2139481 0.02472032 0.0007896108 0.02015600 0.0003859801
## 122   122 0.03227641 0.2141974 0.02471753 0.0007848716 0.01999103 0.0003869209
## 123   123 0.03227859 0.2141153 0.02472194 0.0007824965 0.01996990 0.0003832333
## 124   124 0.03227739 0.2141801 0.02471985 0.0007865484 0.02015117 0.0003853512
## 125   125 0.03226988 0.2145171 0.02471646 0.0007868318 0.02021949 0.0003832078
## 126   126 0.03227981 0.2141037 0.02472450 0.0007839754 0.01974734 0.0003799442
## 127   127 0.03227668 0.2142629 0.02472476 0.0007770422 0.01977257 0.0003715059
## 128   128 0.03228127 0.2141043 0.02473012 0.0007740786 0.01988308 0.0003699428
## 129   129 0.03228257 0.2140426 0.02473057 0.0007743198 0.02005856 0.0003723835
## 130   130 0.03228115 0.2140942 0.02473109 0.0007746068 0.02017804 0.0003729185
## 131   131 0.03228343 0.2139905 0.02473374 0.0007739370 0.01986336 0.0003749385
## 132   132 0.03228962 0.2137657 0.02473694 0.0007775926 0.01990013 0.0003827151
## 133   133 0.03229008 0.2137700 0.02473461 0.0007818234 0.02031708 0.0003845392
## 134   134 0.03228840 0.2138792 0.02473473 0.0007855359 0.02042269 0.0003891061
## 135   135 0.03229462 0.2136247 0.02474026 0.0007821553 0.02059436 0.0003874865
## 136   136 0.03229480 0.2136320 0.02474016 0.0007912122 0.02085632 0.0003953227
## 137   137 0.03229156 0.2137780 0.02473756 0.0007902473 0.02072796 0.0003990039
## 138   138 0.03228708 0.2139802 0.02473099 0.0007891008 0.02069433 0.0003982005
## 139   139 0.03228763 0.2139680 0.02472786 0.0007948860 0.02079682 0.0004017013
## 140   140 0.03228882 0.2139286 0.02473014 0.0007886465 0.02077376 0.0003930317
## 141   141 0.03229700 0.2135550 0.02473824 0.0007780274 0.02040044 0.0003899584
## 142   142 0.03230359 0.2133037 0.02473868 0.0007798746 0.02059502 0.0003919698
## 143   143 0.03230772 0.2130888 0.02474136 0.0007816932 0.02059172 0.0003872797
## 144   144 0.03230725 0.2130802 0.02474120 0.0007813207 0.02023630 0.0003876276
## 145   145 0.03231029 0.2129471 0.02474116 0.0007795320 0.02001875 0.0003939202
## 146   146 0.03231147 0.2129093 0.02474407 0.0007834038 0.01984737 0.0003921626
## 147   147 0.03231354 0.2128429 0.02474667 0.0007784123 0.01993042 0.0003950415
## 148   148 0.03231701 0.2127089 0.02474938 0.0007749306 0.01985020 0.0003921134
## 149   149 0.03232053 0.2125630 0.02475411 0.0007718220 0.01996978 0.0003854299
## 150   150 0.03232307 0.2124405 0.02475608 0.0007766287 0.02006867 0.0003833535
## 151   151 0.03232597 0.2123260 0.02475721 0.0007730687 0.01982289 0.0003830935
## 152   152 0.03232718 0.2122807 0.02475863 0.0007761854 0.01971293 0.0003810422
## 153   153 0.03232480 0.2123958 0.02475678 0.0007691622 0.01948101 0.0003723604
## 154   154 0.03232780 0.2122667 0.02476071 0.0007741230 0.01960951 0.0003773174
## 155   155 0.03232180 0.2125497 0.02475543 0.0007768100 0.01972016 0.0003774371
## 156   156 0.03232211 0.2125409 0.02475264 0.0007745373 0.01948584 0.0003768117
## 157   157 0.03232019 0.2126225 0.02474860 0.0007806056 0.01963101 0.0003812292
## 158   158 0.03231722 0.2127569 0.02474628 0.0007830220 0.01967356 0.0003825805
## 159   159 0.03231642 0.2127991 0.02474950 0.0007903182 0.01989897 0.0003863968
## 160   160 0.03231809 0.2127176 0.02475053 0.0007876485 0.01983771 0.0003813409
## 161   161 0.03231450 0.2128905 0.02474781 0.0007906014 0.01995722 0.0003865417
## 162   162 0.03231628 0.2128217 0.02475047 0.0007870514 0.01984203 0.0003835790
## 163   163 0.03231652 0.2128236 0.02475212 0.0007864822 0.01981717 0.0003817055
## 164   164 0.03232279 0.2125735 0.02475540 0.0007834748 0.01951396 0.0003782665
## 165   165 0.03232360 0.2125520 0.02475839 0.0007826981 0.01961689 0.0003753845
## 166   166 0.03232495 0.2124910 0.02475766 0.0007825144 0.01948164 0.0003763930
## 167   167 0.03232805 0.2123906 0.02475772 0.0007798184 0.01947377 0.0003725254
## 168   168 0.03232898 0.2123497 0.02476000 0.0007792919 0.01951105 0.0003710904
## 169   169 0.03233230 0.2122082 0.02476224 0.0007778043 0.01934343 0.0003695135
## 170   170 0.03233765 0.2119926 0.02476557 0.0007787126 0.01923285 0.0003724189
## 171   171 0.03233583 0.2120485 0.02476586 0.0007809336 0.01928566 0.0003743559
## 172   172 0.03233719 0.2120029 0.02476697 0.0007792179 0.01917393 0.0003744626
## 173   173 0.03234206 0.2118032 0.02477171 0.0007782846 0.01929067 0.0003736492
## 174   174 0.03233861 0.2119442 0.02477264 0.0007748663 0.01927503 0.0003740469
## 175   175 0.03233576 0.2120447 0.02476981 0.0007765825 0.01941060 0.0003767533
## 176   176 0.03234024 0.2118753 0.02477191 0.0007803193 0.01946965 0.0003768825
## 177   177 0.03234061 0.2118605 0.02477242 0.0007799150 0.01943705 0.0003794650
## 178   178 0.03233847 0.2119516 0.02476794 0.0007766783 0.01936465 0.0003814108
## 179   179 0.03234269 0.2117554 0.02476940 0.0007746035 0.01927426 0.0003784234
## 180   180 0.03234021 0.2118519 0.02476530 0.0007755190 0.01935958 0.0003790306
## 181   181 0.03233873 0.2119263 0.02476331 0.0007752000 0.01933310 0.0003778620
## 182   182 0.03233616 0.2120391 0.02476032 0.0007757483 0.01938528 0.0003769974
## 183   183 0.03233569 0.2120546 0.02475855 0.0007783481 0.01943092 0.0003792255
## 184   184 0.03233753 0.2119835 0.02476113 0.0007797579 0.01957251 0.0003800645
## 185   185 0.03233975 0.2119005 0.02476122 0.0007836676 0.01963304 0.0003828298
## 186   186 0.03233793 0.2119818 0.02475900 0.0007854037 0.01967325 0.0003878073
## 187   187 0.03233988 0.2119070 0.02476301 0.0007853405 0.01971118 0.0003898169
## 188   188 0.03233826 0.2119864 0.02476167 0.0007822318 0.01971645 0.0003878753
## 189   189 0.03233869 0.2119851 0.02476208 0.0007790495 0.01979526 0.0003886246
## 190   190 0.03233913 0.2119830 0.02476145 0.0007769012 0.01981784 0.0003870297
## 191   191 0.03233939 0.2119837 0.02476349 0.0007749667 0.01980069 0.0003842887
## 192   192 0.03233969 0.2119785 0.02476486 0.0007771274 0.01973268 0.0003838299
## 193   193 0.03233626 0.2121277 0.02476251 0.0007785295 0.01980909 0.0003830601
## 194   194 0.03233725 0.2120924 0.02476442 0.0007792637 0.01979151 0.0003821241
## 195   195 0.03233693 0.2121129 0.02476304 0.0007779241 0.01968087 0.0003798517
## 196   196 0.03233752 0.2120947 0.02476334 0.0007777482 0.01970085 0.0003795434
## 197   197 0.03233816 0.2120681 0.02476376 0.0007779097 0.01972879 0.0003815971
## 198   198 0.03233695 0.2121201 0.02476186 0.0007773279 0.01969945 0.0003801957
## 199   199 0.03233941 0.2120270 0.02476304 0.0007784648 0.01983056 0.0003800079
## 200   200 0.03233743 0.2121136 0.02476277 0.0007793964 0.01989832 0.0003809556
## 201   201 0.03233789 0.2120856 0.02476214 0.0007772234 0.01979211 0.0003799004
## 202   202 0.03233544 0.2121909 0.02476084 0.0007756651 0.01976029 0.0003788044
## 203   203 0.03233682 0.2121326 0.02476070 0.0007752283 0.01967479 0.0003780675
## 204   204 0.03233694 0.2121254 0.02476099 0.0007769708 0.01964799 0.0003774508
## 205   205 0.03233585 0.2121728 0.02476185 0.0007776237 0.01975332 0.0003778136
## 206   206 0.03233782 0.2120827 0.02476325 0.0007765189 0.01964787 0.0003767862
## 207   207 0.03233816 0.2120633 0.02476386 0.0007750662 0.01958734 0.0003757045
## 208   208 0.03233899 0.2120310 0.02476360 0.0007752742 0.01962379 0.0003755696
## 209   209 0.03233921 0.2120226 0.02476448 0.0007743627 0.01956316 0.0003747725
## 210   210 0.03233793 0.2120789 0.02476356 0.0007740847 0.01955857 0.0003742106
## 211   211 0.03233886 0.2120399 0.02476434 0.0007736823 0.01958487 0.0003722520
## 212   212 0.03234070 0.2119695 0.02476533 0.0007736860 0.01968851 0.0003711108
## 213   213 0.03234147 0.2119384 0.02476545 0.0007746539 0.01970153 0.0003719989
## 214   214 0.03234261 0.2118922 0.02476702 0.0007742604 0.01967221 0.0003727171
## 215   215 0.03234237 0.2118987 0.02476831 0.0007740693 0.01962663 0.0003724871
## 216   216 0.03234121 0.2119489 0.02476638 0.0007735872 0.01965715 0.0003699556
## 217   217 0.03234040 0.2119927 0.02476534 0.0007723776 0.01970442 0.0003680896
## 218   218 0.03233957 0.2120351 0.02476414 0.0007726798 0.01966459 0.0003670780
## 219   219 0.03233937 0.2120419 0.02476435 0.0007730628 0.01964171 0.0003665800
## 220   220 0.03233985 0.2120194 0.02476507 0.0007732996 0.01961400 0.0003665386
## 221   221 0.03233946 0.2120365 0.02476470 0.0007737284 0.01960376 0.0003663927
## 222   222 0.03233955 0.2120339 0.02476491 0.0007752211 0.01962662 0.0003671327
## 223   223 0.03233933 0.2120434 0.02476507 0.0007744084 0.01962756 0.0003666365
## 224   224 0.03233936 0.2120446 0.02476467 0.0007740312 0.01963260 0.0003650174
## 225   225 0.03233932 0.2120462 0.02476512 0.0007739615 0.01962856 0.0003656598
## 226   226 0.03233936 0.2120489 0.02476585 0.0007749216 0.01966310 0.0003665493
## 227   227 0.03233904 0.2120664 0.02476611 0.0007752368 0.01966431 0.0003672464
## 228   228 0.03233879 0.2120782 0.02476617 0.0007748036 0.01964511 0.0003673891
## 229   229 0.03233760 0.2121265 0.02476520 0.0007753045 0.01963838 0.0003685072
## 230   230 0.03233848 0.2120872 0.02476632 0.0007747648 0.01960698 0.0003689050
## 231   231 0.03233849 0.2120847 0.02476654 0.0007744438 0.01961617 0.0003689440
## 232   232 0.03233907 0.2120604 0.02476699 0.0007746850 0.01961612 0.0003689958
## 233   233 0.03233896 0.2120633 0.02476704 0.0007747891 0.01960100 0.0003693248
## 234   234 0.03233894 0.2120640 0.02476691 0.0007748588 0.01958655 0.0003695165
## 235   235 0.03233926 0.2120501 0.02476706 0.0007746337 0.01958916 0.0003694871
## 236   236 0.03233938 0.2120450 0.02476702 0.0007749339 0.01958872 0.0003693825
## 237   237 0.03233941 0.2120437 0.02476691 0.0007748246 0.01958728 0.0003692516
## 238   238 0.03233954 0.2120388 0.02476691 0.0007749659 0.01958784 0.0003693548
## 239   239 0.03233954 0.2120394 0.02476685 0.0007750095 0.01958734 0.0003694222
## 240   240 0.03233956 0.2120384 0.02476680 0.0007749933 0.01958899 0.0003693663
## [1] "Best Model"
##    nvmax
## 15    15

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## [1] "Coefficients of final model:"
##                  Estimate         2.5 %        97.5 %
## (Intercept)  1.963910e+00  1.947791e+00  1.980030e+00
## x4          -5.353377e-05 -7.102358e-05 -3.604396e-05
## x7           1.149425e-02  1.026902e-02  1.271949e-02
## x8           4.293628e-04  1.427311e-04  7.159945e-04
## x9           3.324791e-03  2.685469e-03  3.964112e-03
## x10          1.337977e-03  7.438113e-04  1.932143e-03
## x11          2.291277e+05  8.705589e+04  3.711996e+05
## x16          9.873956e-04  5.758899e-04  1.398901e-03
## x17          1.719254e-03  1.089382e-03  2.349127e-03
## x21          1.404906e-04  5.914971e-05  2.218316e-04
## stat14      -9.651107e-04 -1.438841e-03 -4.913807e-04
## stat98       3.672725e-03  3.201860e-03  4.143589e-03
## stat100      7.167091e-04  2.372289e-04  1.196189e-03
## stat110     -3.313066e-03 -3.787795e-03 -2.838337e-03
## stat146     -7.552975e-04 -1.235627e-03 -2.749676e-04
## x18.sqrt     2.640688e-02  2.459080e-02  2.822297e-02

Test

if (algo.forward.caret == TRUE){
    test.model(model.forward, data.test
             ,method = 'leapForward',subopt = NULL
             ,formula = formula, feature.names = feature.names, label.names = label.names
             ,id = id
             ,draw.limits = TRUE, transformation = t)
}

## [1] "Summary of predicted values: "
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.032   2.085   2.097   2.097   2.110   2.149 
## [1] "leapForward  Test MSE: 0.000992395717031664"

Backward Elimination with CV

Train

if (algo.backward.caret == TRUE){
  set.seed(1)
  returned = train.caret.glmselect(formula = formula
                                   ,data =  data.train
                                   ,method = "leapBackward"
                                   ,feature.names =  feature.names)
  model.backward = returned$model
  id = returned$id
}

## Aggregating results
## Selecting tuning parameters
## Fitting nvmax = 15 on full training set
## [1] "All models results"
##     nvmax       RMSE  Rsquared        MAE       RMSESD RsquaredSD        MAESD
## 1       1 0.03426021 0.1102603 0.02678506 0.0007528041 0.02129231 0.0004213658
## 2       2 0.03341255 0.1548136 0.02608465 0.0009549896 0.03477939 0.0005441361
## 3       3 0.03278951 0.1857579 0.02547103 0.0009056890 0.03288428 0.0004986653
## 4       4 0.03227713 0.2113517 0.02470874 0.0008645806 0.03734399 0.0005939377
## 5       5 0.03198524 0.2249611 0.02451264 0.0009072163 0.03436803 0.0005425191
## 6       6 0.03196756 0.2257863 0.02450055 0.0009147022 0.03323145 0.0005522053
## 7       7 0.03186275 0.2306455 0.02443907 0.0008841335 0.03173780 0.0005201270
## 8       8 0.03180485 0.2333632 0.02440347 0.0008909229 0.03190063 0.0004926743
## 9       9 0.03174677 0.2360753 0.02438467 0.0009075872 0.03063354 0.0004965026
## 10     10 0.03168841 0.2387674 0.02432430 0.0008999805 0.02990088 0.0004865291
## 11     11 0.03171484 0.2374598 0.02436301 0.0008726894 0.02848580 0.0004481966
## 12     12 0.03171113 0.2375547 0.02437469 0.0008600439 0.02677835 0.0004351152
## 13     13 0.03167366 0.2392646 0.02432044 0.0008373043 0.02565854 0.0004341691
## 14     14 0.03166420 0.2397724 0.02430440 0.0008508276 0.02621653 0.0004402137
## 15     15 0.03163949 0.2409469 0.02429082 0.0008593315 0.02618453 0.0004442920
## 16     16 0.03164473 0.2406580 0.02428619 0.0008393763 0.02519951 0.0004486484
## 17     17 0.03164325 0.2407726 0.02427935 0.0008673947 0.02562879 0.0004463349
## 18     18 0.03164729 0.2405747 0.02429372 0.0008485749 0.02464523 0.0004330501
## 19     19 0.03168841 0.2386146 0.02432800 0.0008264120 0.02355733 0.0004210316
## 20     20 0.03171110 0.2375644 0.02434183 0.0008271752 0.02314372 0.0004230877
## 21     21 0.03173476 0.2365096 0.02436339 0.0008313945 0.02379484 0.0004333460
## 22     22 0.03175516 0.2355267 0.02439250 0.0008338561 0.02314498 0.0004237346
## 23     23 0.03176916 0.2349201 0.02440245 0.0008327500 0.02340274 0.0004204588
## 24     24 0.03177493 0.2347050 0.02441004 0.0008409963 0.02345496 0.0004257067
## 25     25 0.03178287 0.2343685 0.02441062 0.0008308875 0.02322537 0.0004134362
## 26     26 0.03179619 0.2337741 0.02441438 0.0008324826 0.02359294 0.0004231466
## 27     27 0.03180761 0.2332725 0.02443488 0.0008366801 0.02383342 0.0004217597
## 28     28 0.03182861 0.2323548 0.02443680 0.0008191934 0.02423088 0.0004206251
## 29     29 0.03183561 0.2320413 0.02444476 0.0008105384 0.02410549 0.0004082149
## 30     30 0.03185024 0.2313653 0.02444863 0.0008014904 0.02384830 0.0003856504
## 31     31 0.03185283 0.2312528 0.02444723 0.0007969173 0.02329775 0.0003918088
## 32     32 0.03186868 0.2305862 0.02445835 0.0007905404 0.02396391 0.0003921945
## 33     33 0.03188780 0.2297477 0.02446460 0.0007865565 0.02494340 0.0003973655
## 34     34 0.03189182 0.2295659 0.02447159 0.0007800004 0.02445720 0.0003936162
## 35     35 0.03189635 0.2294232 0.02447373 0.0007769604 0.02419877 0.0003929271
## 36     36 0.03189436 0.2295445 0.02447257 0.0007825188 0.02422240 0.0004090803
## 37     37 0.03190115 0.2292838 0.02447025 0.0007962640 0.02501735 0.0004136543
## 38     38 0.03191729 0.2285613 0.02448650 0.0007798285 0.02474520 0.0004087816
## 39     39 0.03193203 0.2279545 0.02450272 0.0007911723 0.02523623 0.0004108918
## 40     40 0.03194654 0.2273048 0.02451757 0.0008070040 0.02495350 0.0004191922
## 41     41 0.03196409 0.2265445 0.02453187 0.0007934827 0.02489399 0.0004165233
## 42     42 0.03197647 0.2260333 0.02454319 0.0007800708 0.02450155 0.0003986890
## 43     43 0.03199466 0.2252299 0.02455340 0.0007760666 0.02488404 0.0003996308
## 44     44 0.03199553 0.2252195 0.02455370 0.0007800897 0.02426066 0.0003987086
## 45     45 0.03200684 0.2247222 0.02455942 0.0007866806 0.02413003 0.0004041298
## 46     46 0.03201926 0.2241588 0.02456604 0.0008058421 0.02408938 0.0004037689
## 47     47 0.03202895 0.2238202 0.02456513 0.0008116412 0.02421970 0.0004063187
## 48     48 0.03203295 0.2236481 0.02457481 0.0008234586 0.02467805 0.0004161632
## 49     49 0.03204358 0.2231107 0.02458470 0.0008305108 0.02444026 0.0004103283
## 50     50 0.03204396 0.2231292 0.02458094 0.0008357337 0.02463767 0.0004295078
## 51     51 0.03204944 0.2229014 0.02458762 0.0008345093 0.02471264 0.0004198666
## 52     52 0.03205649 0.2226194 0.02459771 0.0008372992 0.02479030 0.0004143381
## 53     53 0.03206633 0.2221715 0.02461110 0.0008425004 0.02416791 0.0004068308
## 54     54 0.03207086 0.2220109 0.02461044 0.0008455208 0.02406010 0.0004139518
## 55     55 0.03209045 0.2211443 0.02462760 0.0008490833 0.02360976 0.0004138940
## 56     56 0.03209355 0.2210285 0.02462680 0.0008406705 0.02330764 0.0004203771
## 57     57 0.03210284 0.2206265 0.02463840 0.0008396013 0.02309222 0.0004277604
## 58     58 0.03210678 0.2204901 0.02464182 0.0008462618 0.02325108 0.0004281123
## 59     59 0.03211087 0.2203377 0.02464786 0.0008337902 0.02314324 0.0004185129
## 60     60 0.03212955 0.2195623 0.02466962 0.0008401360 0.02382235 0.0004296303
## 61     61 0.03213795 0.2191938 0.02468202 0.0008398019 0.02377118 0.0004253069
## 62     62 0.03213857 0.2191469 0.02468521 0.0008385798 0.02330676 0.0004225637
## 63     63 0.03214133 0.2190488 0.02468236 0.0008326046 0.02322222 0.0004161557
## 64     64 0.03215795 0.2183417 0.02469250 0.0008267838 0.02311866 0.0004076196
## 65     65 0.03216192 0.2182107 0.02469294 0.0008255965 0.02299553 0.0004043456
## 66     66 0.03214799 0.2188766 0.02467444 0.0008242654 0.02325083 0.0004064395
## 67     67 0.03214659 0.2189562 0.02466740 0.0008265687 0.02256327 0.0004074907
## 68     68 0.03214538 0.2190359 0.02466483 0.0008332120 0.02217889 0.0004146178
## 69     69 0.03215152 0.2187662 0.02466459 0.0008288934 0.02217275 0.0004056444
## 70     70 0.03214592 0.2190486 0.02465228 0.0008323475 0.02234042 0.0004085425
## 71     71 0.03214800 0.2190132 0.02463969 0.0008270404 0.02216514 0.0004082876
## 72     72 0.03215669 0.2186437 0.02465100 0.0008331795 0.02256193 0.0004147454
## 73     73 0.03215367 0.2187952 0.02464946 0.0008283953 0.02266707 0.0004099737
## 74     74 0.03215676 0.2186839 0.02464656 0.0008280168 0.02276448 0.0004166444
## 75     75 0.03216800 0.2182089 0.02465942 0.0008251146 0.02284854 0.0004205164
## 76     76 0.03217114 0.2181262 0.02466287 0.0008170117 0.02249246 0.0004150882
## 77     77 0.03217437 0.2180012 0.02465861 0.0008186589 0.02252204 0.0004200170
## 78     78 0.03218105 0.2177573 0.02466331 0.0008076583 0.02224158 0.0004185362
## 79     79 0.03219504 0.2172269 0.02467302 0.0008213984 0.02258789 0.0004336141
## 80     80 0.03219841 0.2170796 0.02467804 0.0008251214 0.02289822 0.0004288977
## 81     81 0.03219585 0.2171968 0.02468240 0.0008186735 0.02215611 0.0004144227
## 82     82 0.03220395 0.2168671 0.02468516 0.0008168400 0.02230277 0.0004243540
## 83     83 0.03222304 0.2160394 0.02470337 0.0008181163 0.02203061 0.0004221431
## 84     84 0.03221695 0.2162883 0.02469343 0.0008160227 0.02214504 0.0004252315
## 85     85 0.03220496 0.2168054 0.02468591 0.0008129163 0.02198156 0.0004209713
## 86     86 0.03221559 0.2163283 0.02469734 0.0008149312 0.02199863 0.0004169186
## 87     87 0.03221269 0.2164886 0.02469454 0.0008224955 0.02220621 0.0004188990
## 88     88 0.03222062 0.2161359 0.02469525 0.0008238182 0.02199060 0.0004249613
## 89     89 0.03222288 0.2160442 0.02469390 0.0008185740 0.02211243 0.0004161867
## 90     90 0.03222214 0.2160936 0.02469855 0.0008199664 0.02190726 0.0004229266
## 91     91 0.03222757 0.2158600 0.02469933 0.0008165584 0.02139752 0.0004072534
## 92     92 0.03222759 0.2158808 0.02469633 0.0008157224 0.02125837 0.0004090356
## 93     93 0.03222896 0.2158286 0.02469762 0.0008178038 0.02124651 0.0004123202
## 94     94 0.03222623 0.2159786 0.02469450 0.0008236799 0.02138536 0.0004154254
## 95     95 0.03223989 0.2153981 0.02470226 0.0008161023 0.02090257 0.0004067268
## 96     96 0.03223614 0.2156159 0.02469504 0.0008127834 0.02111494 0.0003988606
## 97     97 0.03222965 0.2159216 0.02469790 0.0008124155 0.02121129 0.0004023824
## 98     98 0.03222647 0.2160474 0.02470231 0.0008095708 0.02115862 0.0003986178
## 99     99 0.03222691 0.2160717 0.02470702 0.0008181810 0.02130720 0.0003985245
## 100   100 0.03223677 0.2156877 0.02471405 0.0008196077 0.02133149 0.0004032755
## 101   101 0.03223593 0.2157674 0.02471611 0.0008153188 0.02147614 0.0004146521
## 102   102 0.03223942 0.2156195 0.02472296 0.0008150733 0.02141926 0.0004164598
## 103   103 0.03223904 0.2156575 0.02472045 0.0008114829 0.02148960 0.0004236423
## 104   104 0.03223530 0.2157903 0.02471805 0.0008037586 0.02110610 0.0004227198
## 105   105 0.03223714 0.2157067 0.02471072 0.0007989645 0.02064543 0.0004159000
## 106   106 0.03224091 0.2155593 0.02471505 0.0007953940 0.02091938 0.0004069479
## 107   107 0.03224115 0.2155315 0.02471475 0.0007883926 0.02082095 0.0004043583
## 108   108 0.03224963 0.2151795 0.02471841 0.0007859827 0.02087297 0.0004103059
## 109   109 0.03224150 0.2155517 0.02471180 0.0007803654 0.02105676 0.0004156144
## 110   110 0.03224807 0.2152743 0.02471472 0.0007841279 0.02081732 0.0004158234
## 111   111 0.03226127 0.2147428 0.02472254 0.0007876026 0.02085142 0.0004213654
## 112   112 0.03225963 0.2148179 0.02471901 0.0007838718 0.02069491 0.0004180054
## 113   113 0.03226603 0.2145443 0.02471765 0.0007919620 0.02057552 0.0004199706
## 114   114 0.03226207 0.2147422 0.02471211 0.0007903734 0.02082164 0.0004143053
## 115   115 0.03226587 0.2145890 0.02471300 0.0007906578 0.02074922 0.0004116694
## 116   116 0.03226900 0.2144841 0.02471028 0.0007939644 0.02062369 0.0004081307
## 117   117 0.03227299 0.2143127 0.02471580 0.0007951192 0.02057980 0.0004054444
## 118   118 0.03226812 0.2145047 0.02471142 0.0007888751 0.02034675 0.0004046123
## 119   119 0.03227017 0.2144148 0.02470817 0.0007899478 0.02014245 0.0003997822
## 120   120 0.03226998 0.2143998 0.02470961 0.0007845679 0.01974533 0.0003917600
## 121   121 0.03228581 0.2137869 0.02472333 0.0007908302 0.02024558 0.0003883268
## 122   122 0.03227883 0.2140971 0.02471963 0.0007876860 0.02016197 0.0003900738
## 123   123 0.03228385 0.2139087 0.02472217 0.0007771444 0.02004026 0.0003816897
## 124   124 0.03227676 0.2142409 0.02471913 0.0007864519 0.02024998 0.0003903156
## 125   125 0.03227221 0.2144342 0.02471769 0.0007823018 0.02021296 0.0003857499
## 126   126 0.03227998 0.2141133 0.02472443 0.0007780603 0.01989196 0.0003837035
## 127   127 0.03227727 0.2142413 0.02472841 0.0007792930 0.02004828 0.0003796874
## 128   128 0.03228079 0.2140966 0.02473466 0.0007830192 0.02027311 0.0003847867
## 129   129 0.03228359 0.2139677 0.02473331 0.0007833062 0.02029018 0.0003819154
## 130   130 0.03228984 0.2137296 0.02473777 0.0007896304 0.02040915 0.0003944726
## 131   131 0.03228794 0.2138000 0.02473663 0.0007799018 0.01998886 0.0003858585
## 132   132 0.03229269 0.2136181 0.02473696 0.0007800528 0.02004915 0.0003879103
## 133   133 0.03229383 0.2136224 0.02473808 0.0007844891 0.02042456 0.0003892024
## 134   134 0.03229410 0.2136440 0.02474010 0.0007856311 0.02057084 0.0003909050
## 135   135 0.03229672 0.2135407 0.02474164 0.0007889407 0.02074911 0.0003978830
## 136   136 0.03229211 0.2137606 0.02473849 0.0007912750 0.02093370 0.0003992342
## 137   137 0.03229019 0.2138309 0.02473255 0.0007912184 0.02081773 0.0004056778
## 138   138 0.03229006 0.2138463 0.02473012 0.0007868169 0.02053301 0.0003995370
## 139   139 0.03228993 0.2138655 0.02473099 0.0007929037 0.02073171 0.0003987834
## 140   140 0.03229295 0.2137530 0.02473165 0.0007897114 0.02096085 0.0003908186
## 141   141 0.03230392 0.2132728 0.02473794 0.0007821096 0.02067601 0.0003872118
## 142   142 0.03229978 0.2134559 0.02473500 0.0007876005 0.02063485 0.0003968913
## 143   143 0.03230264 0.2132873 0.02473611 0.0007881398 0.02055559 0.0003934592
## 144   144 0.03230922 0.2129977 0.02474082 0.0007845742 0.02030030 0.0003961371
## 145   145 0.03231024 0.2129393 0.02474155 0.0007778682 0.02002050 0.0003968329
## 146   146 0.03231348 0.2128216 0.02474998 0.0007731435 0.01973545 0.0003936839
## 147   147 0.03231824 0.2126225 0.02475263 0.0007713733 0.01975078 0.0003889733
## 148   148 0.03232421 0.2123780 0.02475518 0.0007715684 0.01975498 0.0003913077
## 149   149 0.03232164 0.2124678 0.02475504 0.0007786404 0.01984642 0.0003918223
## 150   150 0.03232341 0.2124152 0.02475321 0.0007784563 0.01980779 0.0003854496
## 151   151 0.03232250 0.2124679 0.02475735 0.0007755318 0.01973154 0.0003899907
## 152   152 0.03232340 0.2124386 0.02476056 0.0007758978 0.01972476 0.0003850567
## 153   153 0.03232601 0.2123290 0.02475885 0.0007816456 0.01971407 0.0003829650
## 154   154 0.03232358 0.2124208 0.02475399 0.0007790325 0.01965950 0.0003748283
## 155   155 0.03231738 0.2126993 0.02474538 0.0007838250 0.01969428 0.0003754770
## 156   156 0.03231566 0.2127792 0.02474461 0.0007815400 0.01934242 0.0003727855
## 157   157 0.03232176 0.2125167 0.02475097 0.0007781833 0.01928881 0.0003729557
## 158   158 0.03231541 0.2128088 0.02474578 0.0007800208 0.01952148 0.0003751752
## 159   159 0.03231252 0.2129431 0.02474522 0.0007855872 0.01972332 0.0003793809
## 160   160 0.03231342 0.2129182 0.02474597 0.0007832039 0.01974340 0.0003785855
## 161   161 0.03231785 0.2127288 0.02474841 0.0007890005 0.01961327 0.0003860520
## 162   162 0.03232130 0.2125981 0.02475254 0.0007902053 0.01957500 0.0003868808
## 163   163 0.03232328 0.2125303 0.02475429 0.0007923755 0.01967038 0.0003841165
## 164   164 0.03232661 0.2124154 0.02475586 0.0007898913 0.01963139 0.0003801395
## 165   165 0.03232735 0.2123675 0.02476103 0.0007890430 0.01959355 0.0003799296
## 166   166 0.03232542 0.2124570 0.02476035 0.0007857457 0.01944702 0.0003782942
## 167   167 0.03232668 0.2124257 0.02475974 0.0007831935 0.01939707 0.0003747444
## 168   168 0.03232830 0.2123689 0.02475897 0.0007830951 0.01938223 0.0003727361
## 169   169 0.03233531 0.2120738 0.02476242 0.0007800110 0.01926721 0.0003693912
## 170   170 0.03233898 0.2119301 0.02476435 0.0007795049 0.01915034 0.0003705776
## 171   171 0.03233935 0.2118858 0.02476795 0.0007806651 0.01918395 0.0003741178
## 172   172 0.03233787 0.2119802 0.02476830 0.0007791931 0.01928464 0.0003746573
## 173   173 0.03234196 0.2118103 0.02477218 0.0007815114 0.01943568 0.0003741463
## 174   174 0.03233891 0.2119309 0.02477328 0.0007771914 0.01944670 0.0003743095
## 175   175 0.03233663 0.2120195 0.02476800 0.0007781725 0.01942572 0.0003762304
## 176   176 0.03233992 0.2118997 0.02477200 0.0007798896 0.01942896 0.0003769328
## 177   177 0.03233822 0.2119690 0.02477014 0.0007783717 0.01945596 0.0003792569
## 178   178 0.03234082 0.2118552 0.02476927 0.0007766635 0.01954599 0.0003830463
## 179   179 0.03234123 0.2118270 0.02476923 0.0007769566 0.01939478 0.0003777740
## 180   180 0.03233604 0.2120356 0.02476323 0.0007790026 0.01938293 0.0003780981
## 181   181 0.03233477 0.2121116 0.02475974 0.0007788330 0.01943100 0.0003759096
## 182   182 0.03233661 0.2120389 0.02476091 0.0007788404 0.01948532 0.0003778095
## 183   183 0.03233805 0.2119756 0.02476209 0.0007798531 0.01953488 0.0003798196
## 184   184 0.03234024 0.2118879 0.02476323 0.0007819545 0.01972025 0.0003812310
## 185   185 0.03233925 0.2119249 0.02476006 0.0007846495 0.01962482 0.0003824769
## 186   186 0.03233878 0.2119479 0.02475800 0.0007862196 0.01971964 0.0003875352
## 187   187 0.03233988 0.2119070 0.02476301 0.0007853405 0.01971118 0.0003898169
## 188   188 0.03233826 0.2119864 0.02476167 0.0007822318 0.01971645 0.0003878753
## 189   189 0.03233869 0.2119851 0.02476208 0.0007790495 0.01979526 0.0003886246
## 190   190 0.03233878 0.2119936 0.02476119 0.0007778321 0.01979143 0.0003870488
## 191   191 0.03233601 0.2121162 0.02476120 0.0007756188 0.01964909 0.0003848292
## 192   192 0.03233932 0.2119813 0.02476415 0.0007774001 0.01972553 0.0003842260
## 193   193 0.03233693 0.2120958 0.02476280 0.0007791111 0.01983366 0.0003829389
## 194   194 0.03233795 0.2120619 0.02476496 0.0007798593 0.01981442 0.0003818863
## 195   195 0.03233664 0.2121282 0.02476264 0.0007776706 0.01966919 0.0003800219
## 196   196 0.03233716 0.2121042 0.02476298 0.0007773025 0.01967624 0.0003784888
## 197   197 0.03233956 0.2120110 0.02476495 0.0007770292 0.01975243 0.0003798745
## 198   198 0.03233761 0.2120966 0.02476298 0.0007766258 0.01976250 0.0003790415
## 199   199 0.03233926 0.2120381 0.02476327 0.0007782155 0.01986980 0.0003800747
## 200   200 0.03233858 0.2120665 0.02476247 0.0007793616 0.01982402 0.0003805686
## 201   201 0.03233833 0.2120648 0.02476182 0.0007774234 0.01973778 0.0003797530
## 202   202 0.03233658 0.2121409 0.02476098 0.0007757364 0.01970815 0.0003784589
## 203   203 0.03233725 0.2121165 0.02476076 0.0007750473 0.01969490 0.0003780707
## 204   204 0.03233819 0.2120761 0.02476247 0.0007763502 0.01970829 0.0003775549
## 205   205 0.03233659 0.2121419 0.02476230 0.0007768303 0.01976253 0.0003779411
## 206   206 0.03233782 0.2120827 0.02476325 0.0007765189 0.01964787 0.0003767862
## 207   207 0.03233816 0.2120633 0.02476386 0.0007750662 0.01958734 0.0003757045
## 208   208 0.03233899 0.2120310 0.02476360 0.0007752742 0.01962379 0.0003755696
## 209   209 0.03233974 0.2119998 0.02476558 0.0007737821 0.01956989 0.0003750736
## 210   210 0.03233844 0.2120568 0.02476407 0.0007735364 0.01956523 0.0003743565
## 211   211 0.03233886 0.2120399 0.02476434 0.0007736823 0.01958487 0.0003722520
## 212   212 0.03234025 0.2119892 0.02476487 0.0007730586 0.01965520 0.0003707717
## 213   213 0.03234065 0.2119714 0.02476536 0.0007737219 0.01965907 0.0003715822
## 214   214 0.03234225 0.2119060 0.02476729 0.0007739540 0.01966213 0.0003725923
## 215   215 0.03234288 0.2118850 0.02476799 0.0007738735 0.01964904 0.0003723367
## 216   216 0.03234188 0.2119303 0.02476663 0.0007732229 0.01968127 0.0003699761
## 217   217 0.03234052 0.2119937 0.02476534 0.0007722439 0.01970410 0.0003680921
## 218   218 0.03233972 0.2120305 0.02476467 0.0007725122 0.01966596 0.0003672435
## 219   219 0.03233932 0.2120445 0.02476443 0.0007729789 0.01963969 0.0003667033
## 220   220 0.03233980 0.2120221 0.02476513 0.0007732091 0.01961187 0.0003666285
## 221   221 0.03233946 0.2120365 0.02476470 0.0007737284 0.01960376 0.0003663927
## 222   222 0.03233968 0.2120298 0.02476509 0.0007753340 0.01962974 0.0003672881
## 223   223 0.03233946 0.2120410 0.02476524 0.0007745236 0.01963246 0.0003667865
## 224   224 0.03233949 0.2120405 0.02476484 0.0007741442 0.01963567 0.0003651737
## 225   225 0.03233932 0.2120462 0.02476512 0.0007739615 0.01962856 0.0003656598
## 226   226 0.03233936 0.2120489 0.02476585 0.0007749216 0.01966310 0.0003665493
## 227   227 0.03233904 0.2120664 0.02476611 0.0007752368 0.01966431 0.0003672464
## 228   228 0.03233879 0.2120782 0.02476617 0.0007748036 0.01964511 0.0003673891
## 229   229 0.03233760 0.2121265 0.02476520 0.0007753045 0.01963838 0.0003685072
## 230   230 0.03233848 0.2120872 0.02476632 0.0007747648 0.01960698 0.0003689050
## 231   231 0.03233849 0.2120847 0.02476654 0.0007744438 0.01961617 0.0003689440
## 232   232 0.03233907 0.2120604 0.02476699 0.0007746850 0.01961612 0.0003689958
## 233   233 0.03233896 0.2120633 0.02476704 0.0007747891 0.01960100 0.0003693248
## 234   234 0.03233894 0.2120640 0.02476691 0.0007748588 0.01958655 0.0003695165
## 235   235 0.03233926 0.2120501 0.02476706 0.0007746337 0.01958916 0.0003694871
## 236   236 0.03233938 0.2120450 0.02476702 0.0007749339 0.01958872 0.0003693825
## 237   237 0.03233941 0.2120437 0.02476691 0.0007748246 0.01958728 0.0003692516
## 238   238 0.03233954 0.2120388 0.02476691 0.0007749659 0.01958784 0.0003693548
## 239   239 0.03233954 0.2120394 0.02476685 0.0007750095 0.01958734 0.0003694222
## 240   240 0.03233956 0.2120384 0.02476680 0.0007749933 0.01958899 0.0003693663
## [1] "Best Model"
##    nvmax
## 15    15

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## [1] "Coefficients of final model:"
##                  Estimate         2.5 %        97.5 %
## (Intercept)  1.963910e+00  1.947791e+00  1.980030e+00
## x4          -5.353377e-05 -7.102358e-05 -3.604396e-05
## x7           1.149425e-02  1.026902e-02  1.271949e-02
## x8           4.293628e-04  1.427311e-04  7.159945e-04
## x9           3.324791e-03  2.685469e-03  3.964112e-03
## x10          1.337977e-03  7.438113e-04  1.932143e-03
## x11          2.291277e+05  8.705589e+04  3.711996e+05
## x16          9.873956e-04  5.758899e-04  1.398901e-03
## x17          1.719254e-03  1.089382e-03  2.349127e-03
## x21          1.404906e-04  5.914971e-05  2.218316e-04
## stat14      -9.651107e-04 -1.438841e-03 -4.913807e-04
## stat98       3.672725e-03  3.201860e-03  4.143589e-03
## stat100      7.167091e-04  2.372289e-04  1.196189e-03
## stat110     -3.313066e-03 -3.787795e-03 -2.838337e-03
## stat146     -7.552975e-04 -1.235627e-03 -2.749676e-04
## x18.sqrt     2.640688e-02  2.459080e-02  2.822297e-02

Test

if (algo.backward.caret == TRUE){
  test.model(model.backward, data.test
             ,method = 'leapBackward',subopt = NULL
             ,formula = formula, feature.names = feature.names, label.names = label.names
             ,id = id
             ,draw.limits = TRUE, transformation = t)
}

## [1] "Summary of predicted values: "
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.032   2.085   2.097   2.097   2.110   2.149 
## [1] "leapBackward  Test MSE: 0.000992395717031665"

Stepwise Selection with CV

Train

if (algo.stepwise.caret == TRUE){
  set.seed(1)
  returned = train.caret.glmselect(formula = formula
                                   ,data =  data.train
                                   ,method = "leapSeq"
                                   ,feature.names = feature.names)
  model.stepwise = returned$model
  id = returned$id
}

## Aggregating results
## Selecting tuning parameters
## Fitting nvmax = 18 on full training set
## [1] "All models results"
##     nvmax       RMSE  Rsquared        MAE       RMSESD RsquaredSD        MAESD
## 1       1 0.03426021 0.1102603 0.02678506 0.0007528041 0.02129231 0.0004213658
## 2       2 0.03341255 0.1548136 0.02608465 0.0009549896 0.03477939 0.0005441361
## 3       3 0.03278951 0.1857579 0.02547103 0.0009056890 0.03288428 0.0004986653
## 4       4 0.03227713 0.2113517 0.02470874 0.0008645806 0.03734399 0.0005939377
## 5       5 0.03198524 0.2249611 0.02451264 0.0009072163 0.03436803 0.0005425191
## 6       6 0.03196756 0.2257863 0.02450055 0.0009147022 0.03323145 0.0005522053
## 7       7 0.03186275 0.2306455 0.02443907 0.0008841335 0.03173780 0.0005201270
## 8       8 0.03180485 0.2333632 0.02440347 0.0008909229 0.03190063 0.0004926743
## 9       9 0.03174677 0.2360753 0.02438467 0.0009075872 0.03063354 0.0004965026
## 10     10 0.03168841 0.2387674 0.02432430 0.0008999805 0.02990088 0.0004865291
## 11     11 0.03169907 0.2381655 0.02434296 0.0008523372 0.02756473 0.0004382474
## 12     12 0.03171113 0.2375547 0.02437469 0.0008600439 0.02677835 0.0004351152
## 13     13 0.03200490 0.2226034 0.02465558 0.0015671846 0.06681294 0.0012041844
## 14     14 0.03166420 0.2397724 0.02430440 0.0008508276 0.02621653 0.0004402137
## 15     15 0.03163949 0.2409469 0.02429082 0.0008593315 0.02618453 0.0004442920
## 16     16 0.03165164 0.2403430 0.02429781 0.0008321322 0.02526408 0.0004478658
## 17     17 0.03165851 0.2400597 0.02429486 0.0008510357 0.02571485 0.0004432393
## 18     18 0.03163761 0.2410311 0.02428919 0.0008637303 0.02482473 0.0004450328
## 19     19 0.03197312 0.2235867 0.02457625 0.0010062941 0.05516598 0.0008752393
## 20     20 0.03197949 0.2233679 0.02455323 0.0007454009 0.04829339 0.0004830157
## 21     21 0.03173476 0.2365096 0.02436339 0.0008313945 0.02379484 0.0004333460
## 22     22 0.03207813 0.2191086 0.02462715 0.0012410605 0.05560380 0.0008145365
## 23     23 0.03199136 0.2232666 0.02457199 0.0010264215 0.05081490 0.0006865414
## 24     24 0.03179461 0.2338135 0.02443553 0.0008431137 0.02463055 0.0004305614
## 25     25 0.03211150 0.2179468 0.02475889 0.0015164303 0.06348317 0.0011722005
## 26     26 0.03221701 0.2130780 0.02479872 0.0014641065 0.05078950 0.0010579248
## 27     27 0.03254169 0.1970330 0.02509424 0.0020046793 0.08399034 0.0015117859
## 28     28 0.03204653 0.2208775 0.02458339 0.0010147819 0.05023934 0.0006740563
## 29     29 0.03213600 0.2172938 0.02466175 0.0012303719 0.04623564 0.0009051064
## 30     30 0.03185880 0.2309478 0.02445404 0.0008050190 0.02355782 0.0004140711
## 31     31 0.03255911 0.1964859 0.02501215 0.0019301170 0.07381269 0.0014170684
## 32     32 0.03212428 0.2169001 0.02466072 0.0006895764 0.04567243 0.0004610042
## 33     33 0.03220379 0.2138838 0.02480333 0.0014706052 0.06240923 0.0011672366
## 34     34 0.03219455 0.2146988 0.02469640 0.0011895860 0.04543765 0.0008999051
## 35     35 0.03220277 0.2137855 0.02470765 0.0011586550 0.05356909 0.0007337005
## 36     36 0.03189655 0.2294505 0.02447863 0.0007776587 0.02417592 0.0003962460
## 37     37 0.03247505 0.2000156 0.02493592 0.0012233887 0.06216365 0.0011193100
## 38     38 0.03212955 0.2174295 0.02464398 0.0009451173 0.04812882 0.0006310883
## 39     39 0.03193991 0.2275755 0.02451044 0.0007860591 0.02475719 0.0004125949
## 40     40 0.03279218 0.1845538 0.02520984 0.0015595446 0.07505150 0.0012251982
## 41     41 0.03297090 0.1763319 0.02534927 0.0018918078 0.07233713 0.0013448120
## 42     42 0.03226772 0.2117584 0.02475508 0.0011518058 0.04379280 0.0008795168
## 43     43 0.03237681 0.2064568 0.02485839 0.0014978001 0.06003528 0.0010975392
## 44     44 0.03198733 0.2255056 0.02455071 0.0007911748 0.02407855 0.0003831792
## 45     45 0.03269079 0.1908050 0.02512984 0.0018473723 0.06592762 0.0013367278
## 46     46 0.03258664 0.1944599 0.02512349 0.0014798619 0.07429498 0.0012964680
## 47     47 0.03280184 0.1857338 0.02522154 0.0018265261 0.06766167 0.0013456653
## 48     48 0.03240412 0.2053495 0.02487957 0.0015082994 0.05907608 0.0010770392
## 49     49 0.03306443 0.1722415 0.02548384 0.0018870053 0.07797897 0.0013405949
## 50     50 0.03231960 0.2090136 0.02490663 0.0014235201 0.05912435 0.0011156091
## 51     51 0.03223824 0.2127324 0.02471612 0.0009529229 0.04624799 0.0006011677
## 52     52 0.03269420 0.1907087 0.02520216 0.0018366052 0.07573378 0.0014375256
## 53     53 0.03243561 0.2040980 0.02490065 0.0015107592 0.05835313 0.0010870623
## 54     54 0.03206740 0.2221724 0.02459821 0.0008375350 0.02384293 0.0004094759
## 55     55 0.03258365 0.1955333 0.02497689 0.0011663929 0.05983910 0.0007728523
## 56     56 0.03274163 0.1886948 0.02523539 0.0018417423 0.07509640 0.0014429258
## 57     57 0.03258818 0.1947715 0.02498901 0.0013691681 0.06271333 0.0009846771
## 58     58 0.03248161 0.2022654 0.02493994 0.0015162427 0.05819785 0.0011142514
## 59     59 0.03262598 0.1921229 0.02502533 0.0008687430 0.06513372 0.0008551064
## 60     60 0.03234406 0.2070991 0.02479318 0.0007612227 0.04675897 0.0004432228
## 61     61 0.03239393 0.2063494 0.02485972 0.0011924040 0.04401910 0.0008771546
## 62     62 0.03264875 0.1920569 0.02501805 0.0010672001 0.05569341 0.0008411652
## 63     63 0.03298012 0.1763828 0.02543633 0.0017095108 0.07193430 0.0013237338
## 64     64 0.03244520 0.2037523 0.02488181 0.0010985446 0.04838627 0.0006467292
## 65     65 0.03238378 0.2054057 0.02481080 0.0007346983 0.04570533 0.0004117181
## 66     66 0.03335947 0.1571987 0.02584053 0.0019664703 0.08721492 0.0015311205
## 67     67 0.03267983 0.1905481 0.02503200 0.0011062002 0.05768050 0.0008720779
## 68     68 0.03251663 0.2007399 0.02500186 0.0014686752 0.05134131 0.0010346187
## 69     69 0.03292308 0.1789223 0.02530331 0.0015784799 0.07703464 0.0011256445
## 70     70 0.03256637 0.1954343 0.02491202 0.0008032307 0.05747340 0.0005690655
## 71     71 0.03254543 0.1995308 0.02500215 0.0014654666 0.05105647 0.0010436991
## 72     72 0.03266215 0.1912635 0.02509910 0.0012998122 0.06641083 0.0010147541
## 73     73 0.03263868 0.1936076 0.02507851 0.0015135988 0.07019458 0.0011064970
## 74     74 0.03269599 0.1896959 0.02501854 0.0009502313 0.05783864 0.0005993396
## 75     75 0.03216933 0.2181565 0.02466650 0.0008371304 0.02288286 0.0004264993
## 76     76 0.03247948 0.2026533 0.02489938 0.0012389126 0.04682898 0.0009391196
## 77     77 0.03253924 0.2000993 0.02496354 0.0014734364 0.05657768 0.0011169470
## 78     78 0.03236338 0.2076091 0.02478868 0.0009211083 0.04492778 0.0006141751
## 79     79 0.03285701 0.1843417 0.02527825 0.0018443851 0.07626370 0.0013896952
## 80     80 0.03279920 0.1868887 0.02520890 0.0016849908 0.07048863 0.0012799010
## 81     81 0.03371680 0.1397710 0.02591576 0.0016123702 0.07480412 0.0013564416
## 82     82 0.03311640 0.1694614 0.02539313 0.0014088150 0.06614297 0.0010217000
## 83     83 0.03247785 0.2022275 0.02490950 0.0009362179 0.05125990 0.0008480185
## 84     84 0.03304077 0.1742065 0.02541946 0.0017529953 0.07708364 0.0013807392
## 85     85 0.03220692 0.2167420 0.02469054 0.0008017377 0.02183547 0.0004135169
## 86     86 0.03221508 0.2163657 0.02470543 0.0008019339 0.02178429 0.0004046968
## 87     87 0.03222149 0.2161042 0.02470887 0.0008191402 0.02211702 0.0004163802
## 88     88 0.03279820 0.1865528 0.02521165 0.0015372411 0.07472505 0.0012214912
## 89     89 0.03323389 0.1632888 0.02559192 0.0017463308 0.08303527 0.0014516929
## 90     90 0.03251770 0.2012134 0.02492155 0.0012163649 0.04593042 0.0009202539
## 91     91 0.03340325 0.1561680 0.02574804 0.0017804000 0.07756671 0.0013182787
## 92     92 0.03221946 0.2162595 0.02469021 0.0008243319 0.02158716 0.0004168150
## 93     93 0.03302336 0.1733501 0.02533779 0.0013942600 0.07947318 0.0011782834
## 94     94 0.03222983 0.2157896 0.02470660 0.0008287097 0.02132207 0.0004073701
## 95     95 0.03289306 0.1827537 0.02522324 0.0015523832 0.06498826 0.0011159597
## 96     96 0.03293556 0.1813974 0.02535792 0.0019118611 0.07500112 0.0013889920
## 97     97 0.03246402 0.2022768 0.02485139 0.0007639436 0.04634773 0.0004568322
## 98     98 0.03252429 0.2008413 0.02493926 0.0010750435 0.04713801 0.0006273854
## 99     99 0.03280327 0.1873779 0.02523611 0.0019067541 0.07561378 0.0013751561
## 100   100 0.03246248 0.2023956 0.02485764 0.0007633733 0.04622087 0.0004480062
## 101   101 0.03254378 0.2005523 0.02500363 0.0015061454 0.06186956 0.0010420025
## 102   102 0.03264950 0.1958557 0.02509464 0.0015246153 0.05425124 0.0011209119
## 103   103 0.03254503 0.2005299 0.02500657 0.0015068723 0.06203048 0.0010477680
## 104   104 0.03288968 0.1835950 0.02527077 0.0018837940 0.07096742 0.0014329345
## 105   105 0.03243559 0.2046713 0.02485292 0.0009404413 0.04633102 0.0006474864
## 106   106 0.03255054 0.2003227 0.02499796 0.0015106145 0.06218653 0.0010588459
## 107   107 0.03244588 0.2042427 0.02486009 0.0009351503 0.04648182 0.0006410827
## 108   108 0.03224900 0.2152140 0.02471380 0.0007920549 0.02094642 0.0004137188
## 109   109 0.03244695 0.2041920 0.02486047 0.0009437589 0.04655217 0.0006493135
## 110   110 0.03357546 0.1486055 0.02581837 0.0020009994 0.08502649 0.0015360109
## 111   111 0.03286891 0.1839594 0.02529617 0.0014788597 0.06623330 0.0010647275
## 112   112 0.03323185 0.1662339 0.02557556 0.0018870485 0.07770921 0.0015458780
## 113   113 0.03286578 0.1850184 0.02526490 0.0019239645 0.07618745 0.0014043353
## 114   114 0.03280926 0.1851060 0.02512701 0.0009270755 0.05829049 0.0005961029
## 115   115 0.03226681 0.2145424 0.02471225 0.0007916689 0.02064924 0.0004119487
## 116   116 0.03327290 0.1636512 0.02553013 0.0017515797 0.07469717 0.0014018809
## 117   117 0.03277678 0.1861284 0.02510491 0.0008395112 0.06381350 0.0008687698
## 118   118 0.03256160 0.1999976 0.02498529 0.0014885259 0.05322118 0.0010880768
## 119   119 0.03264519 0.1963022 0.02500761 0.0014496528 0.05514634 0.0010798893
## 120   120 0.03226804 0.2144763 0.02471012 0.0007868033 0.01972932 0.0003907377
## 121   121 0.03269678 0.1917941 0.02512118 0.0012989212 0.05344484 0.0010110089
## 122   122 0.03279612 0.1881949 0.02520303 0.0013130199 0.05277743 0.0011031751
## 123   123 0.03243115 0.2047506 0.02486390 0.0006669425 0.03520638 0.0004330723
## 124   124 0.03281738 0.1874196 0.02512332 0.0011777548 0.04916489 0.0008353239
## 125   125 0.03245523 0.2043000 0.02489021 0.0007888071 0.03873722 0.0006894282
## 126   126 0.03257862 0.1992888 0.02496322 0.0010232176 0.04352704 0.0006069843
## 127   127 0.03227936 0.2141487 0.02473016 0.0007772435 0.01994059 0.0003762790
## 128   128 0.03227953 0.2141572 0.02473061 0.0007714045 0.01999396 0.0003743417
## 129   129 0.03227830 0.2141924 0.02472791 0.0007744807 0.02016826 0.0003740321
## 130   130 0.03228860 0.2137809 0.02473662 0.0007900658 0.02034152 0.0003941740
## 131   131 0.03301961 0.1766699 0.02536761 0.0014856578 0.06514986 0.0010992444
## 132   132 0.03229274 0.2136238 0.02473704 0.0007801196 0.02003909 0.0003879666
## 133   133 0.03258921 0.1967811 0.02499315 0.0007443202 0.04591140 0.0005677860
## 134   134 0.03229410 0.2136440 0.02474010 0.0007856311 0.02057084 0.0003909050
## 135   135 0.03229773 0.2134859 0.02474310 0.0007879773 0.02068380 0.0003956302
## 136   136 0.03261240 0.1963390 0.02504414 0.0011151718 0.04467824 0.0008325959
## 137   137 0.03252213 0.2025624 0.02487032 0.0010373179 0.03450699 0.0006747428
## 138   138 0.03228886 0.2139092 0.02473060 0.0007877162 0.02061861 0.0004021387
## 139   139 0.03256088 0.1989855 0.02505341 0.0010217170 0.04247499 0.0007999031
## 140   140 0.03277623 0.1900521 0.02509640 0.0011278782 0.04306152 0.0006842554
## 141   141 0.03230265 0.2133254 0.02473883 0.0007809932 0.02063646 0.0003867723
## 142   142 0.03246920 0.2044665 0.02488700 0.0008936184 0.04129788 0.0006341345
## 143   143 0.03230216 0.2133205 0.02473715 0.0007890415 0.02064677 0.0003912278
## 144   144 0.03230688 0.2131008 0.02473884 0.0007830151 0.02022017 0.0003966267
## 145   145 0.03293139 0.1820256 0.02518850 0.0012260040 0.04972632 0.0009335254
## 146   146 0.03250568 0.2032790 0.02494296 0.0012258934 0.03953457 0.0008911099
## 147   147 0.03248756 0.2038270 0.02496698 0.0010561372 0.03814764 0.0007394057
## 148   148 0.03275726 0.1883747 0.02517710 0.0010593375 0.04738822 0.0009032608
## 149   149 0.03291900 0.1796423 0.02530868 0.0009728474 0.05771253 0.0008171156
## 150   150 0.03275141 0.1914618 0.02508962 0.0013699790 0.04624983 0.0010152230
## 151   151 0.03297507 0.1799907 0.02533140 0.0012567942 0.04877346 0.0008232400
## 152   152 0.03232623 0.2123224 0.02475943 0.0007741861 0.01973812 0.0003803414
## 153   153 0.03251049 0.2031080 0.02492873 0.0011638983 0.04335194 0.0007334779
## 154   154 0.03251734 0.2028365 0.02495328 0.0012233988 0.03926107 0.0008678396
## 155   155 0.03264658 0.1949548 0.02504479 0.0010596530 0.04858157 0.0007046614
## 156   156 0.03231853 0.2126711 0.02474864 0.0007823963 0.01934516 0.0003832608
## 157   157 0.03287990 0.1833578 0.02519433 0.0009891159 0.05015231 0.0007466431
## 158   158 0.03260146 0.1995998 0.02498921 0.0012346498 0.03792159 0.0008076680
## 159   159 0.03231542 0.2128369 0.02474674 0.0007840497 0.01979479 0.0003779156
## 160   160 0.03260385 0.1995005 0.02499242 0.0012443942 0.03841945 0.0008081441
## 161   161 0.03251534 0.2029624 0.02494688 0.0012503425 0.04035724 0.0008877029
## 162   162 0.03290476 0.1826845 0.02521566 0.0012945797 0.05028860 0.0009852825
## 163   163 0.03232087 0.2126422 0.02475282 0.0007884861 0.01953833 0.0003817910
## 164   164 0.03232300 0.2125721 0.02475382 0.0007856721 0.01954520 0.0003796819
## 165   165 0.03306946 0.1766407 0.02538798 0.0014866264 0.05629872 0.0009483844
## 166   166 0.03245195 0.2051684 0.02488844 0.0007406907 0.03090354 0.0005676735
## 167   167 0.03259538 0.1992343 0.02497067 0.0009522438 0.03840549 0.0005271611
## 168   168 0.03272168 0.1925954 0.02512958 0.0013159155 0.05336947 0.0010041650
## 169   169 0.03264518 0.1949326 0.02509603 0.0009474110 0.04365013 0.0006916047
## 170   170 0.03257014 0.2007859 0.02492621 0.0010822894 0.03381060 0.0006607604
## 171   171 0.03250088 0.2034485 0.02497122 0.0010526241 0.03702172 0.0007141032
## 172   172 0.03263198 0.1984194 0.02502244 0.0012544149 0.03882618 0.0008353300
## 173   173 0.03234129 0.2118288 0.02477128 0.0007791398 0.01928415 0.0003735436
## 174   174 0.03245146 0.2052766 0.02488887 0.0007317649 0.02992359 0.0005443266
## 175   175 0.03257956 0.2003694 0.02493323 0.0011090793 0.03536732 0.0006772445
## 176   176 0.03268309 0.1941390 0.02502724 0.0009846689 0.03939067 0.0007683524
## 177   177 0.03233963 0.2119021 0.02477123 0.0007786152 0.01936827 0.0003787499
## 178   178 0.03253876 0.2022266 0.02495133 0.0011916805 0.04418171 0.0007547929
## 179   179 0.03283334 0.1885540 0.02523140 0.0015372651 0.04909268 0.0011216441
## 180   180 0.03253778 0.2022475 0.02494960 0.0011941926 0.04418992 0.0007572986
## 181   181 0.03245076 0.2053006 0.02487644 0.0007360293 0.03016821 0.0005495584
## 182   182 0.03281298 0.1892332 0.02507113 0.0012754133 0.04286911 0.0008834036
## 183   183 0.03245045 0.2053276 0.02487352 0.0007394563 0.03018665 0.0005488941
## 184   184 0.03250659 0.2032244 0.02497423 0.0010575832 0.03754238 0.0007248236
## 185   185 0.03233975 0.2119005 0.02476122 0.0007836676 0.01963304 0.0003828298
## 186   186 0.03253421 0.2023374 0.02496341 0.0012501209 0.04037074 0.0009144853
## 187   187 0.03264806 0.1957261 0.02506037 0.0011460465 0.04820389 0.0008273187
## 188   188 0.03233826 0.2119864 0.02476167 0.0007822318 0.01971645 0.0003878753
## 189   189 0.03233869 0.2119851 0.02476208 0.0007790495 0.01979526 0.0003886246
## 190   190 0.03233894 0.2119923 0.02476104 0.0007768696 0.01983093 0.0003872731
## 191   191 0.03233939 0.2119837 0.02476349 0.0007749667 0.01980069 0.0003842887
## 192   192 0.03288869 0.1863463 0.02520388 0.0014420843 0.04578330 0.0009921659
## 193   193 0.03233693 0.2120958 0.02476280 0.0007791111 0.01983366 0.0003829389
## 194   194 0.03251343 0.2030529 0.02497746 0.0010736300 0.03840866 0.0007268128
## 195   195 0.03233664 0.2121282 0.02476264 0.0007776706 0.01966919 0.0003800219
## 196   196 0.03233712 0.2121063 0.02476277 0.0007774062 0.01969201 0.0003797941
## 197   197 0.03266754 0.1941865 0.02511150 0.0009650432 0.04505095 0.0007172576
## 198   198 0.03233820 0.2120682 0.02476267 0.0007769896 0.01975927 0.0003802039
## 199   199 0.03271942 0.1926770 0.02514848 0.0011619461 0.05327973 0.0008631609
## 200   200 0.03262722 0.1982521 0.02499375 0.0009884962 0.04034625 0.0005774699
## 201   201 0.03263095 0.1980946 0.02499582 0.0009928108 0.04055962 0.0005827746
## 202   202 0.03262977 0.1981477 0.02499573 0.0009924009 0.04060020 0.0005824297
## 203   203 0.03264462 0.1980091 0.02503111 0.0012736255 0.03991654 0.0008696827
## 204   204 0.03233804 0.2120777 0.02476241 0.0007765086 0.01970782 0.0003775372
## 205   205 0.03233659 0.2121419 0.02476230 0.0007768303 0.01976253 0.0003779411
## 206   206 0.03266084 0.1948383 0.02505225 0.0008833044 0.04868921 0.0007498102
## 207   207 0.03233816 0.2120633 0.02476386 0.0007750662 0.01958734 0.0003757045
## 208   208 0.03265196 0.1956450 0.02505825 0.0011435912 0.04843800 0.0008170874
## 209   209 0.03233921 0.2120226 0.02476448 0.0007743627 0.01956316 0.0003747725
## 210   210 0.03249264 0.2032119 0.02490248 0.0006680661 0.03356549 0.0004405823
## 211   211 0.03245821 0.2050668 0.02488914 0.0007349337 0.03083145 0.0005659010
## 212   212 0.03234070 0.2119695 0.02476533 0.0007736860 0.01968851 0.0003711108
## 213   213 0.03255296 0.2014111 0.02493741 0.0009534177 0.04491345 0.0006650987
## 214   214 0.03281808 0.1889785 0.02512721 0.0012427853 0.05295144 0.0008874842
## 215   215 0.03252377 0.2026923 0.02498991 0.0010798652 0.03866899 0.0007437722
## 216   216 0.03234116 0.2119540 0.02476628 0.0007736106 0.01965083 0.0003699500
## 217   217 0.03249035 0.2033511 0.02489902 0.0006656094 0.03313424 0.0004271639
## 218   218 0.03255661 0.2013434 0.02494207 0.0009611554 0.04529124 0.0006756475
## 219   219 0.03245915 0.2050601 0.02488948 0.0007348316 0.03090059 0.0005648241
## 220   220 0.03233980 0.2120221 0.02476513 0.0007732091 0.01961187 0.0003666285
## 221   221 0.03290651 0.1830118 0.02529317 0.0012180960 0.05803164 0.0010031310
## 222   222 0.03233955 0.2120339 0.02476491 0.0007752211 0.01962662 0.0003671327
## 223   223 0.03233933 0.2120434 0.02476507 0.0007744084 0.01962756 0.0003666365
## 224   224 0.03233949 0.2120405 0.02476484 0.0007741442 0.01963567 0.0003651737
## 225   225 0.03325016 0.1683168 0.02553365 0.0016489595 0.06047643 0.0011755957
## 226   226 0.03233936 0.2120489 0.02476585 0.0007749216 0.01966310 0.0003665493
## 227   227 0.03249056 0.2034074 0.02490424 0.0006690850 0.03313585 0.0004335597
## 228   228 0.03233879 0.2120782 0.02476617 0.0007748036 0.01964511 0.0003673891
## 229   229 0.03233760 0.2121265 0.02476520 0.0007753045 0.01963838 0.0003685072
## 230   230 0.03255749 0.2016811 0.02496442 0.0012429660 0.04646462 0.0007982236
## 231   231 0.03233849 0.2120847 0.02476654 0.0007744438 0.01961617 0.0003689440
## 232   232 0.03233907 0.2120604 0.02476699 0.0007746850 0.01961612 0.0003689958
## 233   233 0.03281237 0.1891698 0.02518206 0.0012848004 0.04757993 0.0009695255
## 234   234 0.03233894 0.2120640 0.02476691 0.0007748588 0.01958655 0.0003695165
## 235   235 0.03254307 0.2021794 0.02497901 0.0012597917 0.04069534 0.0009118314
## 236   236 0.03345434 0.1576488 0.02577368 0.0015268179 0.05888331 0.0011163873
## 237   237 0.03350972 0.1558862 0.02578852 0.0017672432 0.06198039 0.0012537860
## 238   238 0.03307605 0.1773725 0.02546007 0.0017135446 0.06199986 0.0011647812
## 239   239 0.03348090 0.1557793 0.02566540 0.0010811676 0.05124994 0.0009267201
## 240   240 0.03233956 0.2120384 0.02476680 0.0007749933 0.01958899 0.0003693663
## [1] "Best Model"
##    nvmax
## 18    18

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## [1] "Coefficients of final model:"
##                  Estimate         2.5 %        97.5 %
## (Intercept)  1.963771e+00  1.947652e+00  1.979890e+00
## x4          -5.320052e-05 -7.067149e-05 -3.572956e-05
## x7           1.145572e-02  1.023138e-02  1.268006e-02
## x8           4.151672e-04  1.287736e-04  7.015607e-04
## x9           3.304881e-03  2.666142e-03  3.943619e-03
## x10          1.330109e-03  7.363538e-04  1.923864e-03
## x11          2.322545e+05  9.026901e+04  3.742399e+05
## x16          9.724642e-04  5.613936e-04  1.383535e-03
## x17          1.732465e-03  1.103220e-03  2.361709e-03
## x21          1.441897e-04  6.289987e-05  2.254795e-04
## stat4       -5.620629e-04 -1.042930e-03 -8.119545e-05
## stat13      -6.103082e-04 -1.085991e-03 -1.346250e-04
## stat14      -9.719696e-04 -1.445196e-03 -4.987432e-04
## stat92      -5.379545e-04 -1.017316e-03 -5.859352e-05
## stat98       3.661289e-03  3.190942e-03  4.131635e-03
## stat100      7.073783e-04  2.284089e-04  1.186348e-03
## stat110     -3.314730e-03 -3.788999e-03 -2.840461e-03
## stat146     -7.717429e-04 -1.251591e-03 -2.918949e-04
## x18.sqrt     2.644477e-02  2.463045e-02  2.825908e-02

Test

if (algo.stepwise.caret == TRUE){
  test.model(model.stepwise, data.test
             ,method = 'leapSeq',subopt = NULL
             ,formula = formula, feature.names = feature.names, label.names = label.names
             ,id = id
             ,draw.limits = TRUE, transformation = t)
  
}

## [1] "Summary of predicted values: "
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.033   2.085   2.097   2.097   2.110   2.150 
## [1] "leapSeq  Test MSE: 0.000993509884551637"

LASSO with CV

Train

if (algo.LASSO.caret == TRUE){
  set.seed(1)
  tune.grid= expand.grid(alpha = 1,lambda = 10^seq(from=-4,to=0,length=100))
  returned = train.caret.glmselect(formula = formula
                                   ,data =  data.train
                                   ,method = "glmnet"
                                   ,subopt = 'LASSO'
                                   ,tune.grid = tune.grid
                                   ,feature.names = feature.names)
  model.LASSO.caret = returned$model
}

## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo, : There were missing values in resampled
## performance measures.

## Aggregating results
## Selecting tuning parameters
## Fitting alpha = 1, lambda = 0.000643 on full training set
## glmnet 
## 
## 5584 samples
##  240 predictor
## 
## No pre-processing
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 5026, 5026, 5026, 5025, 5025, 5026, ... 
## Resampling results across tuning parameters:
## 
##   lambda        RMSE        Rsquared   MAE       
##   0.0001000000  0.03210399  0.2203605  0.02460799
##   0.0001097499  0.03208546  0.2210637  0.02459714
##   0.0001204504  0.03206600  0.2218114  0.02458601
##   0.0001321941  0.03204598  0.2225907  0.02457468
##   0.0001450829  0.03202534  0.2234058  0.02456317
##   0.0001592283  0.03200393  0.2242633  0.02455100
##   0.0001747528  0.03198148  0.2251790  0.02453829
##   0.0001917910  0.03195820  0.2261474  0.02452560
##   0.0002104904  0.03193449  0.2271545  0.02451431
##   0.0002310130  0.03191097  0.2281734  0.02450427
##   0.0002535364  0.03188749  0.2292155  0.02449398
##   0.0002782559  0.03186345  0.2303146  0.02448301
##   0.0003053856  0.03183960  0.2314472  0.02447198
##   0.0003351603  0.03181677  0.2325776  0.02446162
##   0.0003678380  0.03179488  0.2337070  0.02445123
##   0.0004037017  0.03177435  0.2348175  0.02444192
##   0.0004430621  0.03175595  0.2358781  0.02443428
##   0.0004862602  0.03174176  0.2367838  0.02443072
##   0.0005336699  0.03173181  0.2375271  0.02443121
##   0.0005857021  0.03172455  0.2381932  0.02443410
##   0.0006428073  0.03172352  0.2386037  0.02444169
##   0.0007054802  0.03172853  0.2387832  0.02445402
##   0.0007742637  0.03173894  0.2387736  0.02446982
##   0.0008497534  0.03175531  0.2385591  0.02449036
##   0.0009326033  0.03177989  0.2380225  0.02452007
##   0.0010235310  0.03181523  0.2370290  0.02455797
##   0.0011233240  0.03186001  0.2356493  0.02460371
##   0.0012328467  0.03191074  0.2340768  0.02465537
##   0.0013530478  0.03196471  0.2324801  0.02471097
##   0.0014849683  0.03202280  0.2308272  0.02476928
##   0.0016297508  0.03208949  0.2288767  0.02483467
##   0.0017886495  0.03216561  0.2265995  0.02490776
##   0.0019630407  0.03224586  0.2243528  0.02498667
##   0.0021544347  0.03232777  0.2223542  0.02507063
##   0.0023644894  0.03241517  0.2204414  0.02516029
##   0.0025950242  0.03250966  0.2186031  0.02525603
##   0.0028480359  0.03261733  0.2165526  0.02536051
##   0.0031257158  0.03274517  0.2138644  0.02547940
##   0.0034304693  0.03289849  0.2101266  0.02561715
##   0.0037649358  0.03308223  0.2048401  0.02577523
##   0.0041320124  0.03329608  0.1977560  0.02595337
##   0.0045348785  0.03352503  0.1899198  0.02614103
##   0.0049770236  0.03377661  0.1804757  0.02634229
##   0.0054622772  0.03407109  0.1666127  0.02657274
##   0.0059948425  0.03438039  0.1501826  0.02680649
##   0.0065793322  0.03466592  0.1347470  0.02701157
##   0.0072208090  0.03493247  0.1202722  0.02719315
##   0.0079248290  0.03515412  0.1114842  0.02734203
##   0.0086974900  0.03534371  0.1102603  0.02746367
##   0.0095454846  0.03556152  0.1102603  0.02760707
##   0.0104761575  0.03582216  0.1102603  0.02778115
##   0.0114975700  0.03613365  0.1102603  0.02799140
##   0.0126185688  0.03629208        NaN  0.02809992
##   0.0138488637  0.03629208        NaN  0.02809992
##   0.0151991108  0.03629208        NaN  0.02809992
##   0.0166810054  0.03629208        NaN  0.02809992
##   0.0183073828  0.03629208        NaN  0.02809992
##   0.0200923300  0.03629208        NaN  0.02809992
##   0.0220513074  0.03629208        NaN  0.02809992
##   0.0242012826  0.03629208        NaN  0.02809992
##   0.0265608778  0.03629208        NaN  0.02809992
##   0.0291505306  0.03629208        NaN  0.02809992
##   0.0319926714  0.03629208        NaN  0.02809992
##   0.0351119173  0.03629208        NaN  0.02809992
##   0.0385352859  0.03629208        NaN  0.02809992
##   0.0422924287  0.03629208        NaN  0.02809992
##   0.0464158883  0.03629208        NaN  0.02809992
##   0.0509413801  0.03629208        NaN  0.02809992
##   0.0559081018  0.03629208        NaN  0.02809992
##   0.0613590727  0.03629208        NaN  0.02809992
##   0.0673415066  0.03629208        NaN  0.02809992
##   0.0739072203  0.03629208        NaN  0.02809992
##   0.0811130831  0.03629208        NaN  0.02809992
##   0.0890215085  0.03629208        NaN  0.02809992
##   0.0977009957  0.03629208        NaN  0.02809992
##   0.1072267222  0.03629208        NaN  0.02809992
##   0.1176811952  0.03629208        NaN  0.02809992
##   0.1291549665  0.03629208        NaN  0.02809992
##   0.1417474163  0.03629208        NaN  0.02809992
##   0.1555676144  0.03629208        NaN  0.02809992
##   0.1707352647  0.03629208        NaN  0.02809992
##   0.1873817423  0.03629208        NaN  0.02809992
##   0.2056512308  0.03629208        NaN  0.02809992
##   0.2257019720  0.03629208        NaN  0.02809992
##   0.2477076356  0.03629208        NaN  0.02809992
##   0.2718588243  0.03629208        NaN  0.02809992
##   0.2983647240  0.03629208        NaN  0.02809992
##   0.3274549163  0.03629208        NaN  0.02809992
##   0.3593813664  0.03629208        NaN  0.02809992
##   0.3944206059  0.03629208        NaN  0.02809992
##   0.4328761281  0.03629208        NaN  0.02809992
##   0.4750810162  0.03629208        NaN  0.02809992
##   0.5214008288  0.03629208        NaN  0.02809992
##   0.5722367659  0.03629208        NaN  0.02809992
##   0.6280291442  0.03629208        NaN  0.02809992
##   0.6892612104  0.03629208        NaN  0.02809992
##   0.7564633276  0.03629208        NaN  0.02809992
##   0.8302175681  0.03629208        NaN  0.02809992
##   0.9111627561  0.03629208        NaN  0.02809992
##   1.0000000000  0.03629208        NaN  0.02809992
## 
## Tuning parameter 'alpha' was held constant at a value of 1
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were alpha = 1 and lambda = 0.0006428073.

##    alpha       lambda
## 21     1 0.0006428073
##     alpha       lambda       RMSE  Rsquared        MAE       RMSESD RsquaredSD        MAESD
## 1       1 0.0001000000 0.03210399 0.2203605 0.02460799 0.0007935202 0.02145382 0.0003832270
## 2       1 0.0001097499 0.03208546 0.2210637 0.02459714 0.0007946932 0.02163842 0.0003836615
## 3       1 0.0001204504 0.03206600 0.2218114 0.02458601 0.0007962810 0.02184184 0.0003843099
## 4       1 0.0001321941 0.03204598 0.2225907 0.02457468 0.0007979939 0.02206900 0.0003853531
## 5       1 0.0001450829 0.03202534 0.2234058 0.02456317 0.0007996128 0.02231289 0.0003871173
## 6       1 0.0001592283 0.03200393 0.2242633 0.02455100 0.0008011079 0.02255857 0.0003883185
## 7       1 0.0001747528 0.03198148 0.2251790 0.02453829 0.0008028799 0.02283171 0.0003891233
## 8       1 0.0001917910 0.03195820 0.2261474 0.02452560 0.0008050571 0.02312546 0.0003893189
## 9       1 0.0002104904 0.03193449 0.2271545 0.02451431 0.0008070306 0.02344663 0.0003899836
## 10      1 0.0002310130 0.03191097 0.2281734 0.02450427 0.0008089887 0.02377454 0.0003917534
## 11      1 0.0002535364 0.03188749 0.2292155 0.02449398 0.0008109379 0.02410214 0.0003937996
## 12      1 0.0002782559 0.03186345 0.2303146 0.02448301 0.0008130628 0.02447112 0.0003976244
## 13      1 0.0003053856 0.03183960 0.2314472 0.02447198 0.0008159159 0.02493461 0.0004032098
## 14      1 0.0003351603 0.03181677 0.2325776 0.02446162 0.0008194352 0.02547903 0.0004090328
## 15      1 0.0003678380 0.03179488 0.2337070 0.02445123 0.0008241119 0.02602710 0.0004147960
## 16      1 0.0004037017 0.03177435 0.2348175 0.02444192 0.0008296260 0.02657489 0.0004200560
## 17      1 0.0004430621 0.03175595 0.2358781 0.02443428 0.0008347870 0.02714575 0.0004249118
## 18      1 0.0004862602 0.03174176 0.2367838 0.02443072 0.0008390173 0.02765790 0.0004306335
## 19      1 0.0005336699 0.03173181 0.2375271 0.02443121 0.0008423223 0.02808730 0.0004377113
## 20      1 0.0005857021 0.03172455 0.2381932 0.02443410 0.0008451508 0.02854276 0.0004436891
## 21      1 0.0006428073 0.03172352 0.2386037 0.02444169 0.0008481771 0.02896474 0.0004466905
## 22      1 0.0007054802 0.03172853 0.2387832 0.02445402 0.0008524801 0.02948467 0.0004492987
## 23      1 0.0007742637 0.03173894 0.2387736 0.02446982 0.0008564537 0.03005899 0.0004504762
## 24      1 0.0008497534 0.03175531 0.2385591 0.02449036 0.0008594646 0.03063964 0.0004516057
## 25      1 0.0009326033 0.03177989 0.2380225 0.02452007 0.0008614168 0.03127946 0.0004530471
## 26      1 0.0010235310 0.03181523 0.2370290 0.02455797 0.0008625583 0.03196639 0.0004545152
## 27      1 0.0011233240 0.03186001 0.2356493 0.02460371 0.0008628448 0.03262000 0.0004550758
## 28      1 0.0012328467 0.03191074 0.2340768 0.02465537 0.0008630417 0.03333725 0.0004549473
## 29      1 0.0013530478 0.03196471 0.2324801 0.02471097 0.0008629382 0.03404463 0.0004556688
## 30      1 0.0014849683 0.03202280 0.2308272 0.02476928 0.0008605413 0.03466240 0.0004575676
## 31      1 0.0016297508 0.03208949 0.2288767 0.02483467 0.0008560611 0.03518199 0.0004581944
## 32      1 0.0017886495 0.03216561 0.2265995 0.02490776 0.0008518197 0.03567584 0.0004586557
## 33      1 0.0019630407 0.03224586 0.2243528 0.02498667 0.0008486062 0.03612997 0.0004591083
## 34      1 0.0021544347 0.03232777 0.2223542 0.02507063 0.0008446877 0.03642562 0.0004554876
## 35      1 0.0023644894 0.03241517 0.2204414 0.02516029 0.0008394507 0.03653857 0.0004492301
## 36      1 0.0025950242 0.03250966 0.2186031 0.02525603 0.0008348327 0.03658424 0.0004415357
## 37      1 0.0028480359 0.03261733 0.2165526 0.02536051 0.0008303861 0.03671172 0.0004328230
## 38      1 0.0031257158 0.03274517 0.2138644 0.02547940 0.0008261283 0.03684089 0.0004248913
## 39      1 0.0034304693 0.03289849 0.2101266 0.02561715 0.0008222337 0.03689677 0.0004165862
## 40      1 0.0037649358 0.03308223 0.2048401 0.02577523 0.0008189187 0.03680612 0.0004105656
## 41      1 0.0041320124 0.03329608 0.1977560 0.02595337 0.0008169034 0.03631422 0.0004047119
## 42      1 0.0045348785 0.03352503 0.1899198 0.02614103 0.0008200889 0.03546705 0.0004007961
## 43      1 0.0049770236 0.03377661 0.1804757 0.02634229 0.0008280877 0.03392950 0.0003995561
## 44      1 0.0054622772 0.03407109 0.1666127 0.02657274 0.0008347953 0.03166086 0.0004032322
## 45      1 0.0059948425 0.03438039 0.1501826 0.02680649 0.0008307000 0.02970814 0.0004091012
## 46      1 0.0065793322 0.03466592 0.1347470 0.02701157 0.0008244113 0.02604380 0.0004136541
## 47      1 0.0072208090 0.03493247 0.1202722 0.02719315 0.0008294691 0.02162587 0.0004192989
## 48      1 0.0079248290 0.03515412 0.1114842 0.02734203 0.0008331495 0.02133947 0.0004203849
## 49      1 0.0086974900 0.03534371 0.1102603 0.02746367 0.0008460459 0.02129231 0.0004265407
## 50      1 0.0095454846 0.03556152 0.1102603 0.02760707 0.0008587567 0.02129231 0.0004380084
## 51      1 0.0104761575 0.03582216 0.1102603 0.02778115 0.0008728291 0.02129231 0.0004548419
## 52      1 0.0114975700 0.03613365 0.1102603 0.02799140 0.0008882768 0.02129231 0.0004734251
## 53      1 0.0126185688 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 54      1 0.0138488637 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 55      1 0.0151991108 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 56      1 0.0166810054 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 57      1 0.0183073828 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 58      1 0.0200923300 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 59      1 0.0220513074 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 60      1 0.0242012826 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 61      1 0.0265608778 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 62      1 0.0291505306 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 63      1 0.0319926714 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 64      1 0.0351119173 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 65      1 0.0385352859 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 66      1 0.0422924287 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 67      1 0.0464158883 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 68      1 0.0509413801 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 69      1 0.0559081018 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 70      1 0.0613590727 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 71      1 0.0673415066 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 72      1 0.0739072203 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 73      1 0.0811130831 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 74      1 0.0890215085 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 75      1 0.0977009957 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 76      1 0.1072267222 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 77      1 0.1176811952 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 78      1 0.1291549665 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 79      1 0.1417474163 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 80      1 0.1555676144 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 81      1 0.1707352647 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 82      1 0.1873817423 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 83      1 0.2056512308 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 84      1 0.2257019720 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 85      1 0.2477076356 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 86      1 0.2718588243 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 87      1 0.2983647240 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 88      1 0.3274549163 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 89      1 0.3593813664 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 90      1 0.3944206059 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 91      1 0.4328761281 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 92      1 0.4750810162 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 93      1 0.5214008288 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 94      1 0.5722367659 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 95      1 0.6280291442 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 96      1 0.6892612104 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 97      1 0.7564633276 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 98      1 0.8302175681 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 99      1 0.9111627561 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720
## 100     1 1.0000000000 0.03629208       NaN 0.02809992 0.0008692293         NA 0.0004701720

## Warning: Removed 48 rows containing missing values (geom_path).

## Warning: Removed 48 rows containing missing values (geom_point).

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## [1] "Coefficients"
##                model.coef
## (Intercept)  1.990104e+00
## x4          -3.938393e-05
## x7           1.042935e-02
## x8           1.894951e-04
## x9           2.826698e-03
## x10          8.732409e-04
## x11          1.195661e+05
## x14         -2.010102e-05
## x16          6.711387e-04
## x17          1.211343e-03
## x19          3.839363e-05
## x21          7.976827e-05
## stat4       -1.658655e-04
## stat8        3.817524e-05
## stat10      -2.307130e-06
## stat13      -2.989942e-04
## stat14      -6.116428e-04
## stat18      -1.785535e-05
## stat22      -8.833802e-05
## stat23       8.911169e-05
## stat24      -5.623264e-05
## stat30       4.033001e-05
## stat41      -3.338599e-05
## stat46       9.728429e-07
## stat60       1.183025e-04
## stat61      -6.817801e-06
## stat82       1.157306e-05
## stat91      -7.335950e-05
## stat92      -1.575420e-04
## stat96      -1.332241e-04
## stat98       3.307608e-03
## stat100      3.439836e-04
## stat110     -2.952876e-03
## stat113     -4.345356e-05
## stat116      5.826270e-06
## stat129      6.897843e-06
## stat131      8.088172e-06
## stat146     -4.053011e-04
## stat147     -1.055210e-04
## stat148     -8.021059e-05
## stat149     -5.039992e-05
## stat156      6.560043e-05
## stat172      2.102640e-05
## stat195      1.211166e-05
## stat202     -1.041845e-04
## stat217      2.821412e-06
## x18.sqrt     2.500270e-02

Test

if (algo.LASSO.caret == TRUE){
  test.model(model.LASSO.caret, data.test
             ,method = 'glmnet',subopt = "LASSO"
             ,formula = formula, feature.names = feature.names, label.names = label.names
             ,draw.limits = TRUE, transformation = t)
}

## [1] "Summary of predicted values: "
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.043   2.086   2.097   2.097   2.109   2.142 
## [1] "glmnet LASSO Test MSE: 0.000981078338776019"

LARS with CV

Train

if (algo.LARS.caret == TRUE){
  set.seed(1)
  returned = train.caret.glmselect(formula = formula
                                   ,data =  data.train
                                   ,method = "lars"
                                   ,subopt = 'NULL'
                                   ,feature.names = feature.names)
  model.LARS.caret = returned$model
}

## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo, : There were missing values in resampled
## performance measures.

## Aggregating results
## Selecting tuning parameters
## Fitting fraction = 0.374 on full training set
## Least Angle Regression 
## 
## 5584 samples
##  240 predictor
## 
## Pre-processing: centered (240), scaled (240) 
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 5026, 5026, 5026, 5025, 5025, 5026, ... 
## Resampling results across tuning parameters:
## 
##   fraction    RMSE        Rsquared   MAE       
##   0.00000000  0.03629208        NaN  0.02809992
##   0.01010101  0.03587241  0.1102603  0.02781449
##   0.02020202  0.03549825  0.1102603  0.02756510
##   0.03030303  0.03517147  0.1103256  0.02735318
##   0.04040404  0.03488924  0.1231906  0.02716510
##   0.05050505  0.03463225  0.1362221  0.02698942
##   0.06060606  0.03439492  0.1492912  0.02681748
##   0.07070707  0.03417240  0.1611945  0.02664809
##   0.08080808  0.03395836  0.1725497  0.02648327
##   0.09090909  0.03375568  0.1816054  0.02632409
##   0.10101010  0.03356520  0.1886756  0.02617191
##   0.11111111  0.03339186  0.1943484  0.02603162
##   0.12121212  0.03322779  0.1999367  0.02589657
##   0.13131313  0.03307075  0.2051440  0.02576516
##   0.14141414  0.03292376  0.2093860  0.02563893
##   0.15151515  0.03278697  0.2128273  0.02551724
##   0.16161616  0.03266051  0.2156073  0.02540135
##   0.17171717  0.03254449  0.2178423  0.02529081
##   0.18181818  0.03244319  0.2198158  0.02519043
##   0.19191919  0.03235666  0.2215832  0.02510082
##   0.20202020  0.03227602  0.2235003  0.02501724
##   0.21212121  0.03220019  0.2255045  0.02494131
##   0.22222222  0.03212964  0.2275905  0.02487350
##   0.23232323  0.03206444  0.2295393  0.02480984
##   0.24242424  0.03200809  0.2312125  0.02475419
##   0.25252525  0.03195886  0.2326215  0.02470398
##   0.26262626  0.03191259  0.2340270  0.02465615
##   0.27272727  0.03186989  0.2353624  0.02461200
##   0.28282828  0.03183100  0.2365740  0.02457255
##   0.29292929  0.03179639  0.2376196  0.02453718
##   0.30303030  0.03177041  0.2383187  0.02450880
##   0.31313131  0.03175203  0.2387042  0.02448724
##   0.32323232  0.03174069  0.2388204  0.02447402
##   0.33333333  0.03173271  0.2388490  0.02446435
##   0.34343434  0.03172717  0.2388270  0.02445517
##   0.35353535  0.03172319  0.2387887  0.02444726
##   0.36363636  0.03172107  0.2386882  0.02444139
##   0.37373737  0.03172088  0.2385183  0.02443750
##   0.38383838  0.03172314  0.2382469  0.02443490
##   0.39393939  0.03172612  0.2379611  0.02443311
##   0.40404040  0.03172975  0.2376580  0.02443282
##   0.41414141  0.03173362  0.2373567  0.02443229
##   0.42424242  0.03173743  0.2370698  0.02443173
##   0.43434343  0.03174151  0.2367762  0.02443158
##   0.44444444  0.03174615  0.2364657  0.02443193
##   0.45454545  0.03175180  0.2361138  0.02443373
##   0.46464646  0.03175791  0.2357482  0.02443606
##   0.47474747  0.03176441  0.2353698  0.02443855
##   0.48484848  0.03177082  0.2350017  0.02444087
##   0.49494949  0.03177766  0.2346217  0.02444350
##   0.50505051  0.03178490  0.2342289  0.02444681
##   0.51515152  0.03179228  0.2338345  0.02445002
##   0.52525253  0.03179993  0.2334324  0.02445363
##   0.53535354  0.03180764  0.2330338  0.02445713
##   0.54545455  0.03181546  0.2326360  0.02446070
##   0.55555556  0.03182333  0.2322405  0.02446414
##   0.56565657  0.03183161  0.2318307  0.02446767
##   0.57575758  0.03184011  0.2314166  0.02447130
##   0.58585859  0.03184891  0.2309946  0.02447536
##   0.59595960  0.03185796  0.2305669  0.02447963
##   0.60606061  0.03186715  0.2301396  0.02448391
##   0.61616162  0.03187660  0.2297045  0.02448838
##   0.62626263  0.03188599  0.2292778  0.02449280
##   0.63636364  0.03189535  0.2288580  0.02449694
##   0.64646465  0.03190462  0.2284467  0.02450088
##   0.65656566  0.03191413  0.2280292  0.02450484
##   0.66666667  0.03192402  0.2275990  0.02450909
##   0.67676768  0.03193416  0.2271614  0.02451362
##   0.68686869  0.03194445  0.2267208  0.02451841
##   0.69696970  0.03195496  0.2262752  0.02452362
##   0.70707071  0.03196548  0.2258345  0.02452904
##   0.71717172  0.03197624  0.2253882  0.02453497
##   0.72727273  0.03198709  0.2249428  0.02454115
##   0.73737374  0.03199790  0.2245031  0.02454718
##   0.74747475  0.03200873  0.2240678  0.02455331
##   0.75757576  0.03201982  0.2236252  0.02455969
##   0.76767677  0.03203111  0.2231776  0.02456596
##   0.77777778  0.03204248  0.2227304  0.02457224
##   0.78787879  0.03205408  0.2222775  0.02457882
##   0.79797980  0.03206612  0.2218097  0.02458556
##   0.80808081  0.03207826  0.2213425  0.02459245
##   0.81818182  0.03209071  0.2208668  0.02459968
##   0.82828283  0.03210320  0.2203939  0.02460710
##   0.83838384  0.03211577  0.2199219  0.02461474
##   0.84848485  0.03212874  0.2194371  0.02462271
##   0.85858586  0.03214198  0.2189458  0.02463086
##   0.86868687  0.03215546  0.2184479  0.02463943
##   0.87878788  0.03216891  0.2179561  0.02464800
##   0.88888889  0.03218229  0.2174719  0.02465669
##   0.89898990  0.03219569  0.2169925  0.02466554
##   0.90909091  0.03220933  0.2165089  0.02467479
##   0.91919192  0.03222312  0.2160229  0.02468438
##   0.92929293  0.03223709  0.2155342  0.02469416
##   0.93939394  0.03225124  0.2150423  0.02470410
##   0.94949495  0.03226560  0.2145458  0.02471429
##   0.95959596  0.03227999  0.2140532  0.02472452
##   0.96969697  0.03229450  0.2135581  0.02473478
##   0.97979798  0.03230927  0.2130577  0.02474526
##   0.98989899  0.03232433  0.2125493  0.02475606
##   1.00000000  0.03233956  0.2120384  0.02476680
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was fraction = 0.3737374.

##     fraction
## 38 0.3737374

## Warning: Removed 1 rows containing missing values (geom_point).

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## [1] "Coefficients"
##            x4            x7            x8            x9           x10           x11           x14           x16 
## -1.890684e-03  7.073069e-03  5.717077e-04  3.685365e-03  1.247871e-03  7.249935e-04 -5.296714e-05  1.375757e-03 
##           x17           x19           x21         stat4         stat8        stat10        stat13        stat14 
##  1.620762e-03  1.307679e-04  8.436611e-04 -3.130447e-04  9.452336e-05 -5.074379e-06 -5.453694e-04 -1.099356e-03 
##        stat18        stat22        stat23        stat24        stat30        stat41        stat60        stat61 
## -5.991246e-05 -1.799253e-04  1.825354e-04 -1.248379e-04  9.499803e-05 -8.389943e-05  2.343882e-04 -2.122117e-05 
##        stat82        stat91        stat92        stat96        stat98       stat100       stat110       stat113 
##  3.883871e-05 -1.518104e-04 -2.979662e-04 -2.612562e-04  5.838975e-03  6.224400e-04 -5.175122e-03 -1.002359e-04 
##       stat116       stat129       stat131       stat146       stat147       stat148       stat149       stat156 
##  1.750602e-05  2.240628e-05  2.665935e-05 -7.230611e-04 -2.096360e-04 -1.680554e-04 -1.151108e-04  1.433593e-04 
##       stat172       stat195       stat202       stat217      x18.sqrt 
##  6.651632e-05  4.021286e-05 -2.031455e-04  6.389420e-06  1.142576e-02

Test

if (algo.LARS.caret == TRUE){
  test.model(model.LARS.caret, data.test
             ,method = 'lars',subopt = NULL
             ,formula = formula, feature.names = feature.names, label.names = label.names
             ,draw.limits = TRUE, transformation = t)
}

## [1] "Summary of predicted values: "
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.043   2.086   2.097   2.097   2.109   2.142 
## [1] "lars  Test MSE: 0.000981247811228031"

Session Info

sessionInfo()

## R version 3.5.2 (2018-12-20)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17763)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_United States.1252  LC_CTYPE=English_United States.1252    LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C                           LC_TIME=English_United States.1252    
## 
## attached base packages:
## [1] parallel  stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] bindrcpp_0.2.2             knitr_1.21                 htmltools_0.3.6            reshape2_1.4.3            
##  [5] lars_1.2                   doParallel_1.0.14          iterators_1.0.10           caret_6.0-81              
##  [9] leaps_3.0                  ggforce_0.1.3              rlist_0.4.6.1              car_3.0-2                 
## [13] carData_3.0-2              bestNormalize_1.3.0        scales_1.0.0               onewaytests_2.0           
## [17] caTools_1.17.1.1           mosaic_1.5.0               mosaicData_0.17.0          ggformula_0.9.1           
## [21] ggstance_0.3.1             lattice_0.20-38            DT_0.5                     ggiraphExtra_0.2.9        
## [25] ggiraph_0.6.0              investr_1.4.0              glmnet_2.0-16              foreach_1.4.4             
## [29] Matrix_1.2-15              MASS_7.3-51.1              PerformanceAnalytics_1.5.2 xts_0.11-2                
## [33] zoo_1.8-4                  forcats_0.3.0              stringr_1.4.0              dplyr_0.7.8               
## [37] purrr_0.3.0                readr_1.3.1                tidyr_0.8.2                tibble_2.0.1              
## [41] ggplot2_3.1.0              tidyverse_1.2.1            usdm_1.1-18                raster_2.8-19             
## [45] sp_1.3-1                   pacman_0.5.0              
## 
## loaded via a namespace (and not attached):
##   [1] readxl_1.2.0       backports_1.1.3    plyr_1.8.4         lazyeval_0.2.1     splines_3.5.2      mycor_0.1.1       
##   [7] crosstalk_1.0.0    leaflet_2.0.2      digest_0.6.18      magrittr_1.5       mosaicCore_0.6.0   openxlsx_4.1.0    
##  [13] recipes_0.1.4      modelr_0.1.3       gower_0.1.2        colorspace_1.4-0   rvest_0.3.2        ggrepel_0.8.0     
##  [19] haven_2.0.0        xfun_0.4           crayon_1.3.4       jsonlite_1.6       bindr_0.1.1        survival_2.43-3   
##  [25] glue_1.3.0         registry_0.5       gtable_0.2.0       ppcor_1.1          ipred_0.9-8        sjmisc_2.7.7      
##  [31] abind_1.4-5        rngtools_1.3.1     bibtex_0.4.2       Rcpp_1.0.0         xtable_1.8-3       units_0.6-2       
##  [37] foreign_0.8-71     stats4_3.5.2       lava_1.6.5         prodlim_2018.04.18 prediction_0.3.6.2 htmlwidgets_1.3   
##  [43] httr_1.4.0         RColorBrewer_1.1-2 pkgconfig_2.0.2    farver_1.1.0       nnet_7.3-12        labeling_0.3      
##  [49] tidyselect_0.2.5   rlang_0.3.1        later_0.8.0        munsell_0.5.0      cellranger_1.1.0   tools_3.5.2       
##  [55] cli_1.0.1          generics_0.0.2     moments_0.14       sjlabelled_1.0.16  broom_0.5.1        evaluate_0.13     
##  [61] ggdendro_0.1-20    yaml_2.2.0         ModelMetrics_1.2.2 zip_1.0.0          nlme_3.1-137       doRNG_1.7.1       
##  [67] mime_0.6           xml2_1.2.0         compiler_3.5.2     rstudioapi_0.9.0   curl_3.3           tweenr_1.0.1      
##  [73] stringi_1.3.1      highr_0.7          gdtools_0.1.7      stringdist_0.9.5.1 pillar_1.3.1       data.table_1.12.0 
##  [79] bitops_1.0-6       httpuv_1.4.5.1     R6_2.4.0           promises_1.0.1     gridExtra_2.3      rio_0.5.16        
##  [85] codetools_0.2-15   assertthat_0.2.0   pkgmaker_0.27      withr_2.1.2        nortest_1.0-4      mgcv_1.8-26       
##  [91] hms_0.4.2          rpart_4.1-13       quadprog_1.5-5     grid_3.5.2         timeDate_3043.102  class_7.3-14      
##  [97] rmarkdown_1.11     snakecase_0.9.2    shiny_1.2.0        lubridate_1.7.4